Depth and Stanley Depth of the Edge Ideals of r-Fold Bristled Graphs of Some Graphs

Abstract

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

1. Introduction

Let $\mathsf{F}:=K[x_1,x_2,\dots ,x_{\upsilon }]$ be a polynomial ring over a field K with standard grading, that is, $deg\left(x_i\right)=1$, for all i. Let M be a finitely generated graded $\mathsf{F}$-module. Suppose that M admits the following minimal free resolution:

$$0⟶\underset{j\in \mathbb{Z}}{⨁}\mathsf{F}{(-j)}^{{\beta }_{p,j}\left(M\right)}⟶\underset{j\in \mathbb{Z}}{⨁}\mathsf{F}{(-j)}^{{\beta }_{p-1,j}\left(M\right)}⟶\cdots ⟶\underset{j\in \mathbb{Z}}{⨁}\mathsf{F}{(-j)}^{{\beta }_{0,j}\left(M\right)}⟶M⟶0.$$

The projective dimension of M is defined as $pdim\left(M\right)=max\{i:{\beta }_{i,j}\left(M\right)\ne 0\}$. The $depth$ of M is defined to be the common length of all maximal M-sequences in the unique graded maximal ideal $(x_1,x_2,\dots ,x_{\upsilon })$. Let M be a finitely generated ${\mathbb{Z}}^{\upsilon }$-graded $\mathsf{F}$-module. For a homogeneous element $u\in M$ and a subset $A\subset \{x_1,x_2,\dots ,x_n\}$, $uK\left[A\right]$ denotes the K-subspace of M generated by all homogeneous elements of the form $uv$, where v is a monomial in $K\left[A\right]$. The K-subspace, $uK\left[A\right]$, is called a Stanley space of dimension $\left|A\right|$ if it is a free $K\left[A\right]$-module, where $\left|A\right|$ denotes the number of indeterminates in A. A Stanley decomposition $\mathcal{D}$ of M is a presentation of the K-vector space M as a finite direct sum of Stanley spaces:

$$\mathcal{D}:M=\underset{i=1}{\stackrel{s}{⨁}}{a}_{i}K\left[A_i\right].$$

The Stanley depth of decomposition $\mathcal{D}$ is defined as $sdepth\left(\mathcal{D}\right)=min\left\{\right|A_i|:i=1,2,\dots ,s\}$. The Stanley depth of M is defined as

$$sdepth\left(M\right)=max\{sdepth(\mathcal{D}):\mathcal{D}\mathrm{is}\mathrm{a}\mathrm{Stanley}\mathrm{decomposition}\mathrm{of}M\}.$$

Stanley conjectured in [1] that $sdepth\left(M\right)\ge depth\left(M\right)$; this conjecture was later disproved by Duval et al. [2] in 2016. However, it is still important to prove Stanley’s inequality for some special classes of ideals. Herzog et al. gave a method in [3] to compute the Stanley depth of modules of the form $I/J$, where $J\subset I$ are monomial ideals. But in general, it is still too hard to compute Stanley depth even using their method. For further details, we refer the reader to [4,5,6].

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph, where $V\left(G\right)=\{x_1,x_2,\dots ,x_{\upsilon }\}$ is the vertex set and $E\left(G\right)$ is the edge set of graph G. All graphs considered in this paper are simple and undirected. The edge ideal $I\left(G\right)$ of the graph G is the ideal generated by all monomials of the form $x_i{x}_j$ such that $\{x_i,x_j\}\in E\left(G\right)$. In the last decade, the study of edge ideals has gained considerable attention. Various findings on these ideals have demonstrated how combinatorial and algebraic aspects interact; see, for instance, [7,8]. The algebraic invariant depth, Stanley depth, and projective dimension have significant importance in the field of commutative algebra. Establishing the relationship of these invariants with other invariants of commutative algebra and invariants of graph theory are current trends in research.

In general, the invariant depth, Stanley depth, and projective dimension are hard to compute. There are very few classes of ideals for which the formulae of these invariants are known; see, for instance, [4,9,10]. We prove that when we consider the r-fold graph of a ladder graph, circular ladder graph, some king’s graphs, and some circular king’s graphs, then the value of depth, Stanley depth, and projective dimension of the quotients rings of the edge ideals of these graphs are functions of r. We also prove that Stanley’s inequality also holds for these quotient rings. Furthermore, our results give strong motivation for further studies in this direction. For our main results, see Theorem 3, Corollary 4, Theorem 4, Theorem 6, Corollary 4, and Theorem 7.

2. Preliminaries

In this section, we will recall some definitions and notations from graph theory. For terminology and definitions from graph theory, we refer the reader to [11,12,13,14]. Some known results related to depth and Stanley depth are also given in this section. If I is a monomial ideal then $\mathcal{G}\left(I\right)$ denotes its unique minimal set of monomial generators. If u is a monomial of $\mathsf{F}$, then $supp\left(u\right):=\{x_i:x_i|u\}$, and for a monomial ideal I, we define $supp\left(I\right):=\{x_i:x_i|u,\mathrm{for}\mathrm{some}u\in \mathcal{G}\left(I\right)\}$. The degree of a vertex $x_i$ denoted by $deg\left(x_i\right)$ is the number of edges that are incident to $x_i$. Let $\upsilon \ge 1$, a path of length $\upsilon -1$, denoted by $P_{\upsilon }$, be a graph with $V\left(P_{\upsilon }\right)=\{x_1,x_2,\dots ,x_{\upsilon }\}$ and $E\left(P_{\upsilon }\right)=\{\{x_i,x_{i+1}\}:1\le i<\upsilon \}$ (if $\upsilon =1$, then $E\left(P_1\right)=\varnothing$). Let $\upsilon \ge 3$, a cycle of length $\upsilon$ denoted by ${\mathcal{C}}_{\upsilon }$, be a graph with $V\left({\mathcal{C}}_{\upsilon }\right)=\{x_1,x_2,\dots ,x_{\upsilon }\}$ and $E\left({\mathcal{C}}_{\upsilon }\right)=\{\{x_i,x_{i+1}\}:1\le i<\upsilon \}\bigcup \left\{\{x_1,x_{\upsilon }\}\right\}$. A graph is said to be a tree if it is acyclic. A vertex $x_i$ is called a pendant vertex if $deg\left(x_i\right)=1$. For $r\ge 2$, an r-star denoted by ${\mathcal{S}}_r$ is a tree with $(r-1)$ leaves and a single vertex with degree $r-1$. A caterpillar is a tree in which the removal of all pendants leaves a path.

Definition 1

([15]). Let G be a graph and $r\ge 1$ be an integer. The graph obtained by attaching r pendant vertices to each vertex of G is called the r-fold bristled graph of G. The r-fold bristled graph of G is denoted by $Br{s}_r\left(G\right)$.

Definition 2

([16]). The Cartesian product $G_1\square G_2$ of graphs $G_1$ and $G_2$ is a graph with vertex set $V\left(G_1\right)\times V\left(G_2\right)$ and $\{(t_1,u_1),(t_2,u_2)\}\in E(G_1\square G_2)$, whenever

1. 

$\{t_1,t_2\}\in E\left(G_1\right)$ and $u_1=u_2$;

2. 

$t_1=t_2$ and $\{u_1,u_2\}\in E\left(G_2\right)$.

Definition 3

([16]). The strong product $G_1⊠G_2$ of graphs $G_1$ and $G_2$ is a graph with vertex set $V\left(G_1\right)\times V\left(G_2\right)$ and $\{(t_1,u_1),(t_2,u_2)\}\in E(G_1⊠G_2)$, whenever

1. 

$\{t_1,t_2\}\in E\left(G_1\right)$ and $u_1=u_2$;

2. 

$t_1=t_2$ and $\{u_1,u_2\}\in E\left(G_2\right)$;

3. 

$\{t_1,t_2\}\in E\left(G_1\right)$ and $\{u_1,u_2\}\in E\left(G_2\right)$.

Here we introduce some notations that will be used throughout the paper. For $\upsilon \ge 1$, let $D_{\upsilon }:=P_{\upsilon }\square P_2$ and $L_{\upsilon }:=P_{\upsilon }⊠P_2$ be graphs. The graph $D_{\upsilon }$ is known as a ladder graph, whereas the graph $L_{\upsilon }$ is called ($\upsilon \times 2$)-king’s graph. See Figure 1 for examples of $D_{\upsilon }$ and $L_{\upsilon }$. For $\upsilon \ge 3$, let $H_{\upsilon }:={\mathcal{C}}_{\upsilon }\square P_2$ and $T_{\upsilon }:={\mathcal{C}}_{\upsilon }⊠P_2$; the graph $H_{\upsilon }$ is called a circular ladder graph. We define the graph $T_{\upsilon }$ as circular ($\upsilon \times 2$)-king’s graph.

Now we recall some known results that are frequently used in this paper. The following lemma, which is also known as the Depth Lemma, has a crucial role in all proofs of our results concerning depth.

Lemma 1

([17]). If $0\to U\to M\to N\to 0$ is a short exact sequence of modules over a local ring $\mathsf{F}$, or a Noetherian graded ring with ${\mathsf{F}}_0$ local, then

1. 

$depth\left(M\right)\ge min\{depth(N),depth(U\left)\right\}$.

2. 

$depth\left(U\right)\ge min\{depth(M),depth(N)+1\}$.

3. 

$depth\left(N\right)\ge min\{depth(U)-1,depth(M\left)\right\}$.

A similar result for Stanley depth as given in the subsequent lemma is proved by Rauf.

Lemma 2

([18]). Let $0\to U\to V\to W\to 0$ be a short exact sequence of ${\mathbb{Z}}^{\upsilon }$-graded $\mathsf{F}$-module. Then $sdepth\left(V\right)\ge min\{sdepth\left(U\right),sdepth\left(W\right)\}.$

Here is a list of some preliminary lemmas that are referred to many times in the proofs of our results.

Lemma 3

([3]). Let $I\subset \mathsf{F}$ be a monomial ideal. If ${\mathsf{F}}^{\prime }=\mathsf{F}{\otimes }_{K}K\left[x_{\upsilon +1}\right]\cong \mathsf{F}\left[x_{\upsilon +1}\right],$ then $depth({\mathsf{F}}^{\prime }/I{\mathsf{F}}^{\prime })=depth(\mathsf{F}/I)+1$ and $sdepth({\mathsf{F}}^{\prime }/I{\mathsf{F}}^{\prime })=sdepth(\mathsf{F}/I)+1.$

Lemma 4

([19]). If $I=I\left({\mathcal{S}}_{\upsilon }\right)\subseteq \mathsf{F}$ is an edge ideal of υ-star, then

$$depth(\mathsf{F}/I)=sdepth(\mathsf{F}/I)=1.$$

Lemma 5

([20]). Let $I\subset {\mathsf{F}}^{\prime }=K[x_1,\dots ,x_r]$, $J\subset {\mathsf{F}}^{\prime \prime }=K[x_{r+1},\dots ,x_{\upsilon }]$ be monomial ideals, where $1\le r<\upsilon$ and $\mathsf{F}={\mathsf{F}}^{\prime }{\otimes }_K{\mathsf{F}}^{\prime \prime }$. Then

$$depth({\mathsf{F}}^{\prime }/I{\otimes }_K{\mathsf{F}}^{\prime \prime }/J)=depth(\mathsf{F}/(I\mathsf{F}+J\mathsf{F}\left)\right))={depth}_{{\mathsf{F}}^{\prime }}({\mathsf{F}}^{\prime }/I)+{depth}_{{\mathsf{F}}^{\prime \prime }}({\mathsf{F}}^{\prime \prime }/J).$$

Lemma 6

([20]). Let $I\subset {\mathsf{F}}^{\prime }=K[x_1,\dots ,x_r]$ and $J\subset {\mathsf{F}}^{\prime }=K[x_{r+1},\dots ,x_{\upsilon }]$ be monomial ideals, where $1\le r<\upsilon$ and $\mathsf{F}={\mathsf{F}}^{\prime }{\otimes }_K{\mathsf{F}}^{\prime \prime }$. Then

$$sdepth({\mathsf{F}}^{\prime }/I{\otimes }_K{\mathsf{F}}^{{}^{\prime \prime }}/J)=sdepth(\mathsf{F}/(I\mathsf{F}+J\mathsf{F}\left)\right))\ge {depth}_{{\mathsf{F}}^{\prime }}({\mathsf{F}}^{\prime }/I)+{depth}_{\mathsf{F}{}^{\prime \prime }}(\mathsf{F}{}^{\prime \prime }/J).$$

The following results are useful in finding upper bounds for depth and Stanley depth.

Corollary 1

([18]). Let $I\subset \mathsf{F}$ be a monomial ideal. Then $depth(\mathsf{F}/(I:u))\ge depth(\mathsf{F}/I)$ for all monomials $u\notin I$.

Proposition 1

([21]). Let $J\subset \mathsf{F}$ be a monomial ideal. Then for all monomials $u\notin J$,

$$sdepth(\mathsf{F}/J)\le sdepth(\mathsf{F}/(J:u\left)\right).$$

Lemma 7

([22]). Let $I\subset \mathsf{F}$ be a squarefree monomial ideal with $supp\left(I\right)=\{x_1,x_2,\dots ,x_{\upsilon }\}$, let $\mu :=x_{i_1}{x}_{i_2}\dots x_{i_q}\in \mathsf{F}/I$, such that $x_h\mu \in I$, for all $x_h\in \{x_1,x_2,\dots ,x_{\upsilon }\}\setminus supp\left(\mu \right)$. Then $sdepth(\mathsf{F}/I)\le q.$

The following result says that once the value of depth of a module is know then one can find its projective dimension.

Theorem 1

([17]). (Auslander–Buchsbaum formula) If $\mathsf{F}$ is a commutative Noetherian local ring and M is a non-zero finitely generated $\mathsf{F}$-module of finite projective dimension, then

$$pdim\left(M\right)+depth\left(M\right)=depth\left(\mathsf{F}\right).$$

For $r\ge 1$ and $\upsilon \ge 1$, if $P_{\upsilon ,r}:=Br{s}_r\left(P_{\upsilon }\right)$, then clearly $P_{\upsilon ,r}$ is a caterpillar and we have the following values for depth and Stanley depth.

Theorem 2

([23]). Let $r\ge 1$ and $\upsilon \ge 1$. Then

$$depth(K\left[V\left(P_{\upsilon ,r}\right)\right]/I\left(P_{\upsilon ,r}\right))=sdepth(K\left[V\left(P_{\upsilon ,r}\right)\right]/I\left(P_{\upsilon ,r}\right))=\lceil \frac{\upsilon -1}{2}\rceil r+\lceil \frac{\upsilon }{2}\rceil .$$

Throughout this paper, we set ${\mathsf{F}}_{\upsilon ,r}:=K[{\bigcup }_{i=1}^{\upsilon }\{x_i,y_i\}\bigcup {\bigcup }_{j=1}^r\{x_{1j},x_{2j}\dots ,x_{\upsilon j},y_{1j},y_{2j},$$\dots ,y_{\upsilon j}\left\}\right]$, where r and $\upsilon$ are positive integers. Also, $|V\left({\mathsf{F}}_{\upsilon ,r}\right)|=2\upsilon (1+r).$

3. Depth and Stanley Depth of r-Fold Bristled Graph of Ladder Graph and Some King’s Graph

In this section, we determine depth, projective dimension, and Stanley depth of the quotient rings associated with edge ideals of r-fold bristled graphs of graphs $D_{\upsilon }$ and $L_{\upsilon }$. See Figure 2a and Figure 3 for 2-fold bristled graph of graphs $D_4$ and $L_4$, respectively. We label the vertices of $Br{s}_r\left(D_{\upsilon }\right)$ and $Br{s}_r\left(L_{\upsilon }\right)$, as shown in Figure 2a and Figure 3, respectively. For $\upsilon ,r\ge 1$, let $I_{\upsilon ,r}:=I\left(Br{s}_r\left(D_{\upsilon }\right)\right)$ and $L_{\upsilon ,r}:=I\left(Br{s}_r\left(L_{\upsilon }\right)\right)$. If $\mathcal{G}\left(I\right)$ denotes the minimal set of monomial generators of the monomial ideal I, using our labeling, we have

$$\mathcal{G}\left(I_{1,r}\right)=\left\{x_1{y}_1\right\}\bigcup \bigcup _{j=1}^r\{x_1{x}_{1j},y_1{y}_{1j}\},$$

and

$$\mathcal{G}\left(L_{1,r}\right)=\left\{x_1{y}_1\right\}\bigcup \bigcup _{j=1}^r\{x_1{x}_{1j},y_1{y}_{1j}\}.$$

If $\upsilon \ge 2$, then we have

$$\mathcal{G}\left(I_{\upsilon ,r}\right)=\bigcup _{i=1}^{\upsilon -1}\{x_i{x}_{i+1},y_i{y}_{i+1}\}\bigcup \bigcup _{i=1}^{\upsilon }\left\{x_i{y}_i\right\}\bigcup \bigcup _{j=1}^r\{y_1{y}_{1j},\dots ,y_{\upsilon }{y}_{\upsilon j},x_1{x}_{1j},\dots ,x_{\upsilon }{x}_{\upsilon j}\},$$

and

$$\begin{array}{c}\hfill \mathcal{G}\left(L_{\upsilon ,r}\right)=\bigcup _{i=1}^{\upsilon -1}\{x_i{x}_{i+1},y_i{y}_{i+1}\}\bigcup \bigcup _{j=1}^r\{x_1{x}_{1j},..,x_{\upsilon }{x}_{\upsilon j},y_1{y}_{1j},..,y_{\upsilon }{y}_{\upsilon j}\}\bigcup \\ \hfill \bigcup _{i=1}^{\upsilon }\left\{x_i{y}_i\right\}\bigcup \{y_1{x}_2,y_{\upsilon }{x}_{\upsilon -1}\}\bigcup \bigcup _{i=2}^{\upsilon -1}\{y_i{x}_{i-1},y_i{x}_{i+1}\}.\end{array}$$

Note that $Br{s}_r\left(D_1\right)\cong Br{s}_r\left(L_1\right)\cong P_{2,r}$ and ${\mathsf{F}}_{1,r}/I_{1,r}\cong {\mathsf{F}}_{1,r}/L_{1,r}\cong K\left[V\left(P_{2,r}\right)\right]/I\left(P_{2,r}\right)$. We also define a modified graph of $Br{s}_r\left(D_{\upsilon }\right)$ denoted by $A_{\upsilon ,r}$ with the set of vertices $V\left(A_{\upsilon ,r}\right)=V\left(Br{s}_r\left(D_{\upsilon }\right)\right)\bigcup \left\{y_{\upsilon +1}\right\}\bigcup \{y_{(\upsilon +1)1},y_{(\upsilon +1)2},\dots ,y_{(\upsilon +1)r}\}$ and $E\left(A_{\upsilon ,r}\right)=E\left(I_{\upsilon ,r}\right)\bigcup$$\left\{\{y_{\upsilon },y_{\upsilon +1}\}\right\}$$\bigcup {\bigcup }_{j=1}^r\{\{y_{\upsilon },y_{(\upsilon +1)1}\},\{y_{\upsilon },y_{(\upsilon +1)2}\},\dots \{y_{\upsilon },y_{(\upsilon +1)r}\}\}$. See Figure 3b for an example of graph $A_{\upsilon ,r}$ and labeling of vertices of this graph. We set ${\mathsf{F}}_{\upsilon ,r}^{*}:={\mathsf{F}}_{\upsilon ,r}[y_{\upsilon +1},y_{(\upsilon +1)1},$$y_{(\upsilon +1)1},\dots ,y_{(\upsilon +1)r}]$ and $I^{*}:=I\left(A_{\upsilon ,r}\right)$. Clearly, $\mathcal{G}\left(I_{\upsilon ,r}^{*}\right)=\mathcal{G}\left(I_{\upsilon ,r}\right)\bigcup \{y_{\upsilon }{y}_{\upsilon +1},y_{\upsilon }{y}_{(\upsilon +1)1},$$y_{\upsilon }{y}_{(\upsilon +1)2},\dots ,y_{\upsilon }{y}_{(\upsilon +1)r}\}$. Note that $A_{\upsilon ,r}\cong P_{3,r}$, ${\mathsf{F}}_{1,r}^{*}/I_{1,r}^{*}=K\left[V\left(P_{3,r}\right)\right]/I\left(P_{3,r}\right)$ and $\left|V\right({\mathsf{F}}_{\upsilon ,r}^{*}\left)\right|$$=(2\upsilon +1)(1+r)$. To determine depth and Stanley depth of ${\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r}$, we shall first determine the depth and Stanley depth of ${\mathsf{F}}_{\upsilon ,r}^{*}/I^{*}$.

Remark 1.

Let I be a squarefree monomial ideal of $\mathsf{F}$ whose monomial generators have degrees of at most 2. We associate a graph $G_I$ to the ideal I with $V\left(G_I\right)=supp\left(I\right)$ and $E\left(G_I\right)=\{\{x_i,x_j\}:x_i{x}_j\in \mathcal{G}\left(I\right)\}$. Let $x_u\in \mathsf{F}$ be a variable of the polynomial ring $\mathsf{F}$ such that $x_u\notin I$. Then $(I:x_u)$ and $(I,x_u)$ are monomial ideals of $\mathsf{F}$ such that $G_{(I:x_u)}$ and $G_{(I,x_u)}$ are subgraphs of $G_I$. See Figure 4a and Figure 4b for graphs $G_{(I_{4,2}^{*}:y_5)}$ and $G_{(L_{3,2},x_3)}$, respectively.

Remark 2.

While proving our results by induction on $\upsilon$, the special cases, say ${\mathsf{F}}_{0,r}/L_{0,r}$ and ${\mathsf{F}}_{0,r}^{*}/I_{0,r}^{*}$, that might appear in the proofs need to be addressed first. We define ${\mathsf{F}}_{0,r}/L_{0,r}:=K$ and ${\mathsf{F}}_{0,r}^{*}/I_{0,r}^{*}:=K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right)$. Thus, we have $depth({\mathsf{F}}_{0,r}/L_{0,r})=sdepth({\mathsf{F}}_{0,r}/L_{0,r})=0$, and by Lemma 4, we have $depth({\mathsf{F}}_{0,r}^{*}/I_{0,r}^{*})=sdepth({\mathsf{F}}_{0,r}^{*}/I_{0,r}^{*})=1$.

Lemma 8.

Let $\upsilon ,r\ge 1$. Then $depth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})=sdepth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})=(r+1)\upsilon +1.$

Proof. 

First we will prove the result for depth. We will prove this by induction on $\upsilon$. We consider the following short exact sequence:

$$\begin{array}{c}\hfill 0⟶{\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon })\stackrel{·y_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*}⟶{\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*},y_{\upsilon })⟶0.\end{array}$$

By the Depth Lemma,

$$depth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})\ge min\{depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon })),depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*},y_{\upsilon }))\}.$$

(1)

If $\upsilon =1$, then by Theorem 2, $depth({\mathsf{F}}_{1,r}^{*}/I_{1,r}^{*})=depth(K\left[V\left(P_{3,r}\right)\right]/I\left(P_{3,r}\right))=r+2$, as required. Let $\upsilon \ge 2$. After renumbering the variables, we have

$${\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -2,r}^{*}/I_{\upsilon -2,r}^{*}\underset{K}{⨂}K[y_{\upsilon },{\cup }_{j=1}^r\{x_{\upsilon j},y_{(\upsilon -1)j},y_{(\upsilon +1)j}\}].$$

Thus, by induction and Lemma 3,

$$depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -2,r}^{*}/I_{\upsilon -2,r}^{*})+3r+1=(r+1)(\upsilon -2)+1+3r+1=(r+1)\upsilon +r.$$

Also,

$${\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*},y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*}\underset{K}{⨂}K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right)\underset{K}{⨂}K[y_{\upsilon 1},y_{\upsilon 2},\dots ,y_{\upsilon r}].$$

By Lemmas 3 and 5,

$$depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*},y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+depth(K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right))+r.$$

Using induction and Lemma 4,

$$depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*},y_{\upsilon }))=(r+1)(\upsilon -1)+1+1+r=(r+1)\upsilon +1.$$

By Equation (1), we have $depth({\mathsf{F}}_{\upsilon ,r}^{*}/\left(I_{\upsilon ,r}^{*}\right))\ge (r+1)\upsilon +1.$ Now we prove the other inequality. We have ${\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon +1})\cong {\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*}{⨂}_{K}K[y_{\upsilon +1},y_{\upsilon 1},y_{\upsilon 2},\dots ,y_{\upsilon r}],$ by Lemma 3, $depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon +1}))=depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+r+1.$ By induction, we have

$$depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon +1}))=(r+1)(\upsilon -1)+1+r+1=(r+1)\upsilon +1.$$

As $y_{\upsilon +1}\notin I_{\upsilon ,r}^{*}$, so by Corollary 1 $depth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})\le depth({\mathsf{F}}_{\upsilon ,r}^{*}/(I_{\upsilon ,r}^{*}:y_{\upsilon +1}))=(r+1)\upsilon +1.$ This completes the proof for depth.

Now we prove the result for Stanley depth. If $\upsilon =1$, then by Theorem 2, $sdepth({\mathsf{F}}_{1,r}^{*}/I_{1,r}^{*})$$=r+2$. For $\upsilon \ge 2$, the required result follows by applying Lemma 2 instead of the Depth Lemma, Lemma 6 instead of Lemma 5, and Proposition 1 instead of Corollary 1. □

Corollary 2.

Let $\upsilon ,r\ge 1$. Then $pdim({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})=r\left(\upsilon +1\right)+\upsilon .$

Proof. 

The required result follows by using Lemma 8 and Theorem 1. □

Now using the previous lemma, we are able to prove one of the main results of this section.

Theorem 3.

Let $\upsilon ,r\ge 1$. Then $depth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})=sdepth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})=(r+1)\upsilon .$

Proof. 

First we will prove the result for depth. Consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon })\stackrel{·x_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r}⟶{\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon })⟶0.$$

When $\upsilon =1$, it is clear from Theorem 2 that $depth({\mathsf{F}}_{1,r}/I_{1,r})=depth(K\left[V\left(P_{2,r}\right)\right]/I\left(P_{2,r}\right))=r+1$. Let $\upsilon \ge 2$. We have ${\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon })\cong ({\mathsf{F}}_{\upsilon -2,r}^{*}/I_{\upsilon -2,r}^{*}){⨂}_{K}K[\left\{x_{\upsilon }\right\}\bigcup {\bigcup }_{j=1}^r\{y_{\upsilon j},x_{(\upsilon -1)j}\}].$ By Lemma 3, we have

$$\begin{array}{cc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -2,r}^{*}/I_{\upsilon -2,r}^{*})+2r+1.\hfill \end{array}$$

By Lemma 8, $depth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))=(r+1)(\upsilon -2)+1+2r+1=(r+1)\upsilon .$ Now clearly $\mathcal{G}(I_{\upsilon ,r},x_{\upsilon })=\{\mathcal{G}\left(I_{\upsilon -1}^{*}\right),x_{\upsilon }\}$ and ${\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*}{⨂}_{K}K[x_{\upsilon 1},x_{\upsilon 2},\dots ,x_{\upsilon r}]$, and using Lemma 3, $depth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+r.$ By Lemma 8,

$$depth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon }))=(r+1)(\upsilon -1)+1+r=(r+1)\upsilon .$$

Applying the Depth Lemma, $depth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})=(r+1)\upsilon .$

Now we prove the result for Stanley depth. If $\upsilon =1$, then by Theorem 2, we have

$$sdepth({\mathsf{F}}_{1,r}/I_{1,r})=sdepth(K\left[V\left(P_{2,r}\right)\right]/I\left(P_{2,r}\right))=r+1.$$

Let $\upsilon \ge 2$. Applying Lemma 2 on the short exact sequence, we obtain

$$sdepth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})\ge min\{sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon })),sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon }))\}.$$

(2)

Proceeding on the same lines as we did for the depth, we obtain $sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))\ge \upsilon (r+1)$ and $sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r},x_{\upsilon }))\ge \upsilon (r+1)$ and by Equation (2), we have $sdepth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})\ge \upsilon (r+1)$. For the other inequality, since $x_{\upsilon }\notin I_{\upsilon ,r}$ and $sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))=sdepth$$({\mathsf{F}}_{\upsilon -2,r}^{*}/I_{\upsilon -2,r}^{*})+2r+1$, by Lemma 8,

$$sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))=(r+1)(\upsilon -2)+1+2r+1=(r+1)\upsilon .$$

By Proposition 1, we have

$$sdepth({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})\le sdepth({\mathsf{F}}_{\upsilon ,r}/(I_{\upsilon ,r}:x_{\upsilon }))=(r+1)\upsilon .$$

This completes the proof for Stanley depth. □

Corollary 3.

Let $\upsilon ,r\ge 1$. Then $pdim({\mathsf{F}}_{\upsilon ,r}/I_{\upsilon ,r})=(r+1)\upsilon .$

Proof. 

The required result follows by using Theorem 3 and Theorem 1. □

Now we find the depth and Stanley depth of ${\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r}$.

Theorem 4.

Let $\upsilon ,r\ge 1$. Then $depth({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})=sdepth({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$

Proof. 

First we will prove the result for depth. We will prove this by induction on $\upsilon$. Consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon })\stackrel{·x_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r}⟶{\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r},x_{\upsilon })⟶0.$$

By the Depth Lemma,

$$depth({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})\ge min\{depth({\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon })),depth({\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r},x_{\upsilon }))\}.$$

(3)

When $\upsilon =1$, it is clear from Theorem 2 that $depth({\mathsf{F}}_{1,r}/L_{1,r})=depth(K\left[V\left(P_{2,r}\right)\right]/I\left(P_{2,r}\right))=r+1$.

Let $\upsilon \ge 2$, ${\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -2,r}/L_{\upsilon -2,r}{⨂}_{K}K[\left\{x_{\upsilon }\right\}\bigcup {\bigcup }_{j=1}^r\{y_{\upsilon j},x_{(\upsilon -1)j},y_{(\upsilon -1)j}\}].$ Using Lemma 3 and induction on $\upsilon$, clearly

$$\begin{array}{c}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -2,r}/L_{\upsilon -2,r})+3r+1=\lfloor \frac{3(\upsilon -2)}{2}\rfloor r+\lceil \frac{\upsilon -2}{2}\rceil +3r+1\end{array}$$

$$=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$$

Now let $J:=(L_{\upsilon ,r},x_{\upsilon })$ and $\mathcal{G}\left(J\right)=\mathcal{G}\left(L_{\upsilon -1,r}\right)\bigcup \{y_{\upsilon }{x}_{\upsilon -1},y_{\upsilon }{y}_{\upsilon -1},x_{\upsilon }\}\bigcup \{y_{\upsilon }{y}_{\upsilon 1},y_{\upsilon }{y}_{\upsilon 2},\dots ,$$y_{\upsilon }{y}_{\upsilon r}\}.$ Consider the following short exact sequence:

$$\begin{array}{c}\hfill 0⟶{\mathsf{F}}_{\upsilon ,r}/(J:y_{\upsilon })\stackrel{·y_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/J⟶{\mathsf{F}}_{\upsilon ,r}/(J,y_{\upsilon })⟶0.\end{array}$$

Again, using the Depth Lemma, we have

$$depth({\mathsf{F}}_{\upsilon ,r}/J)\ge min\{depth({\mathsf{F}}_{\upsilon ,r}/(J:y_{\upsilon })),depth({\mathsf{F}}_{\upsilon ,r}/(J,y_{\upsilon }))\}.$$

(4)

Here ${\mathsf{F}}_{\upsilon ,r}/(J:y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -2,r}/L_{\upsilon -2,r}{⨂}_{K}K[\left\{y_{\upsilon }\right\}\bigcup {\bigcup }_{j=1}^r\{y_{(\upsilon -1)j},x_{(\upsilon -1)j},x_{\upsilon j}\}].$ Using Lemma 3 and induction on $\upsilon$, we have

$$\begin{array}{c}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(J:y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -2,r}/L_{\upsilon -2,r})+3r+1=\lfloor \frac{3(\upsilon -2)}{2}\rfloor r+\lceil \frac{\upsilon -2}{2}\rceil +3r+1\end{array}$$

$$=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$$

As $\mathcal{G}\left((J,y_{\upsilon })\right)=\mathcal{G}\left(L_{\upsilon -1,r}\right)\bigcup \{x_{\upsilon },y_{\upsilon }\}$ and ${\mathsf{F}}_{\upsilon ,r}/(J,y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}/L_{\upsilon -1,r}{⨂}_{K}K[{\bigcup }_{j=1}^r$$\{y_{\upsilon j},x_{\upsilon j}\}].$ By Lemma 3 and induction on $\upsilon$, we obtain

$$\begin{array}{cc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(J,y_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -1,r}/L_{\upsilon -1,r})+2r=\lfloor \frac{3(\upsilon -1)}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil +2r\hfill \end{array}$$

$$=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$$

By Equation (4), we have

$$depth({\mathsf{F}}_{\upsilon ,r}/J)\ge min\{\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil ,\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil \}=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$$

Now by using Equation (3), we obtain

$$depth({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})\ge min\{\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil ,\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil \}=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$$

For upper bound as $x_{\upsilon }\notin {\mathsf{F}}_{\upsilon ,r}$ and $depth({\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon }))=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil$. By Corollary 1, $depth({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})\le depth({\mathsf{F}}_{\upsilon ,r}/(L_{\upsilon ,r}:x_{\upsilon }))=\lfloor \frac{3\upsilon }{2}\rfloor r+\lceil \frac{\upsilon }{2}\rceil .$ This completes the proof for depth. Now we prove the result for Stanley depth. When $\upsilon =1$, it is clear from Theorem 2 that $sdepth({\mathsf{F}}_{1,r}/L_{1,r})=r+1$. For $\upsilon \ge 2$, the required result follows by applying Lemma 2 instead of the Depth Lemma and Proposition 1 instead of Corollary 1. □

Corollary 4.

Let $\upsilon ,r\ge 1$. Then $pdim({\mathsf{F}}_{\upsilon ,r}/L_{\upsilon ,r})=\lceil \frac{\upsilon }{2}\rceil r+\lfloor \frac{3\upsilon }{2}\rfloor .$

Proof. 

The result follows by using Theorem 4 and Theorem 1. □

Example 1.

If $\upsilon =9$ and $r=4$, then by Theorem 4, we have $depth({\mathsf{F}}_{9,4}/L_{9,4})=sdepth({\mathsf{F}}_{9,4}/L_{9,4})$$=\lfloor \frac{3\left(9\right)}{2}\rfloor \left(4\right)+\lceil \frac{9}{2}\rceil =52+5=57$. Also, by Corollary 4, we have $pdim({\mathsf{F}}_{9,4}/L_{9,4})=\lceil \frac{9}{2}\rceil \left(4\right)+\lfloor \frac{3\left(9\right)}{2}\rfloor =20+13=33.$

4. Depth and Stanley Depth of r-Fold Bristled Graph of Circular Ladder Graph and Some Circular King’s Graph

In this section, we determine the depth and Stanley depth of the quotient rings associated with the edge ideal of r-fold bristled graph of circular ladder graph and $T_{\upsilon }$ graph. Figure 5a,b are examples of 2-fold bristled graphs of a circular ladder graph and $T_6$ graph, respectively. For positive integers $r,\upsilon$ such that $r\ge 1$ and $\upsilon \ge 3$, the minimal set of monomial generators of the edge ideal ${\mathfrak{C}}_{\upsilon ,r}=I\left(Br{s}_r\left(H_{\upsilon }\right)\right)$ is given as $\mathcal{G}\left({\mathfrak{C}}_{\upsilon ,r}\right)=\mathcal{G}\left(I_{\upsilon ,r}\right)\bigcup \{x_1{x}_{\upsilon },y_1{y}_{\upsilon }\}$. For $\upsilon \ge 1$, we also define a new graph $A_{\upsilon ,r}^{\prime }$ with $V\left(A_{\upsilon ,r}^{\prime }\right)={\bigcup }_{i=1}^{\upsilon }\{x_i,y_i\}\bigcup \{y_{\upsilon +1},y_{\upsilon +2}\}\bigcup {\bigcup }_{i=1}^r\{x_{1i},\dots ,x_{\upsilon i},y_{1i},\dots ,y_{(\upsilon +2)i}\}$ and

$$\begin{array}{cc}\hfill E\left(A_{\upsilon ,r}^{{}^{\prime }}\right)=& \bigcup _{i=1}^{\upsilon -1}\left\{\{x_i,x_{i+1}\}\right\}\bigcup \bigcup _{i=1}^{\upsilon +1}\left\{\{y_i,y_{i+1}\}\right\}\bigcup \bigcup _{i=1}^{\upsilon }\left\{\{x_i,y_{i+1}\}\right\}\bigcup \hfill \\ & \bigcup _{i=1}^{\upsilon +2}\{\{y_i,y_{i1}\},\{y_i,y_{i2}\},\dots ,\{y_i,y_{ir}\}\}\bigcup \bigcup _{i=1}^{\upsilon }\{\{x_i,x_{i1}\},\{x_i,x_{i2}\},\dots \{x_i,x_{ir}\}\}.\hfill \end{array}$$

See Figure 6 for an example of $A_{\upsilon ,r}^{\prime }$ graph. We set ${\mathsf{F}}_{\upsilon ,r}^{**}:={\mathsf{F}}_{\upsilon ,r}[\{y_{\upsilon +1},y_{\upsilon +2}\}\bigcup {\bigcup }_{j=1}^r\{y_{(\upsilon +1)j},$$y_{(\upsilon +2)j}\left\}\right]$ and $|V\left({\mathsf{F}}_{\upsilon ,r}^{**}\right)|=2(\upsilon +1)(1+r)$. Let $E_{\upsilon ,r}:=I\left(A_{\upsilon ,r}^{{}^{\prime }}\right)$ and $C_{\upsilon ,r}:=I\left(Br{s}_r\left(T_{\upsilon }\right)\right)$. Clearly, $\mathcal{G}\left(C_{\upsilon ,r}\right)=\mathcal{G}\left(L_{\upsilon ,r}\right)\bigcup \{x_1{x}_{\upsilon },y_1{y}_{\upsilon },x_1{y}_{\upsilon },x_{\upsilon }{y}_1\}$.

To determine the depth and Stanley depth of the quotient rings associated with the edge ideal of the r-fold bristled graph of the circular ladder graph, we shall first determine the depth and Stanley depth of the quotient ring associated with the edge ideal of $A_{\upsilon ,r}^{\prime }$ graph. In Figure 7 we give examples of graphs associated to squarefree monomial ideals $(E_{1,2}:y_3)$, $(E_{1,2},y_3)$, $({\mathfrak{C}}_{6,2}:x_6)$ and $({\mathfrak{C}}_{6,2},x_6)$, as discussed in Remark 1. These examples will be helpful in understanding the proofs of our next results.

Remark 3.

While proving our results by induction on υ, we have special case ${\mathsf{F}}_{0,r}^{**}/E_{0,r}$, so we define ${\mathsf{F}}_{0,r}^{**}/E_{0,r}:=K\left[V\left(P_{2,r}\right)\right]/I\left(P_{2,r}\right)$. By using Theorem 2, $depth({\mathsf{F}}_{0,r}^{**}/E_{0,r})=sdepth({\mathsf{F}}_{0,r}^{**}/E_{0,r})=r+1$.

Theorem 5.

Let $r,\upsilon \ge 1$. Then

$$depth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})=sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})=\left\{\begin{array}{cc}(\upsilon +1)(r+1),\hfill & \mathit{if}\upsilon \mathit{is}\mathit{even};\hfill \\ \upsilon (r+1)+2,\hfill & \mathit{if}\upsilon \mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

Proof. 

First we will prove the result for depth by using induction on $\upsilon$. Consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:y_{\upsilon +2})\stackrel{·y_{\upsilon +2}}{\to }{\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r}⟶{\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},y_{\upsilon +2})⟶0.$$

Let $\upsilon =1$. We have ${\mathsf{F}}_{1,r}^{**}/(E_{1,r}:y_3)\cong {⨂}_{i=1}^{2}K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right){⨂}_{K}K[y_3,y_{21},y_{22},\dots ,$$y_{2r}]$, and by Lemmas 3–5, we have

$$depth({\mathsf{F}}_{1,r}^{**}/(E_{1,r}:y_3))=2·depth(K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right))+r+1=2+r+1=r+3.$$

Also, we can see that ${\mathsf{F}}_{1,r}^{**}/(E_{1,r},y_3)\cong K\left[V\left(P_{3,r}\right)\right]/I\left(P_{3,r}\right){⨂}_{K}K[y_{31},y_{32},\dots ,y_{3r}]$. By Lemma 3 and Theorem 2, we have

$$depth({\mathsf{F}}_{1,r}^{**}/(E_{1,r},y_3))=depth(K\left[V\left(P_{3,r}\right)\right]/I\left(P_{3,r}\right))+r=r+2+r=2r+2.$$

Since $depth({\mathsf{F}}_{1,r}^{**}/(E_{1,r}:y_3))\le depth({\mathsf{F}}_{1,r}^{**}/(E_{1,r},y_3))$, then by then Depth Lemma,

$$depth({\mathsf{F}}_{1,r}^{**}/E_{1,r})=depth({\mathsf{F}}_{1,r}^{**}/(E_{1,r}:y_3))=r+3.$$

This prove the result for $\upsilon =1.$

Let $\upsilon \ge 2,$ and $J^{*}:=(E_{\upsilon ,r}:y_{\upsilon +2})$. Now consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon })\stackrel{·x_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}^{**}/J^{*}⟶{\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon })⟶0.$$

We have

$${\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -2,r}^{**}/E_{\upsilon -2,r}\underset{K}{⨂}K[\{x_{\upsilon },y_{\upsilon +2}\}\bigcup \bigcup _{j=1}^r\{y_{(\upsilon +1)j},x_{(\upsilon -1)j}\}],$$

and

$${\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*}\underset{K}{⨂}K[\left\{y_{\upsilon +2}\right\}\bigcup \bigcup _{j=1}^r\{x_{\upsilon j},y_{(\upsilon +1)j}\}].$$

Thus, by using Lemma 3, we obtain $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -2,r}^{**}/E_{\upsilon -2,r})+2r+2$ and $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+2r+1.$ We consider two cases:

Case 1. If $\upsilon$ is even, then by induction on $\upsilon$,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -2,r}^{**}/E_{\upsilon -2,r})+2r+2\hfill & \\ & =(n-2+1)(r+1)+2r+2\hfill & \\ & =\upsilon (r+1)-r-1+2r+2\hfill & \\ & =(\upsilon +1)(r+1).\hfill \end{array}$$

Similarly, by induction on $\upsilon$, we have

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+2r+1\hfill & \\ & =(\upsilon -1)(r+1)+1+2r+1\hfill & \\ & =\upsilon (r+1)-r-1+1+2r+1\hfill & \\ & =(\upsilon +1)(r+1).\hfill \end{array}$$

Since $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon }))$ Applying the Depth Lemma, we obtain

$$depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:y_{\upsilon +2}))=depth({\mathsf{F}}_{\upsilon ,r}^{**}/J^{*})=(\upsilon +1)(r+1).$$

Now ${\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},y_{\upsilon +2})\cong {\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*}{⨂}_{K}K[y_{(\upsilon +2)1},y_{(\upsilon +2)2},\dots ,y_{(\upsilon +2)r}].$ By Lemmas 3 and 8, we have $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},y_{\upsilon +2}))=depth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})+r=(r+1)\upsilon +1+r=(\upsilon +1)(r+1)$. Again, since $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:y_{\upsilon +2}))=depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},y_{\upsilon +2}))$, then by the Depth Lemma,

$$depth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})=(\upsilon +1)(r+1).$$

Case 2. If $\upsilon$ is odd, then by induction on $\upsilon$,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -2,r}^{**}/E_{\upsilon -2,r})+2r+2\hfill & \\ & =(\upsilon -2)(r+1)+2+2r+2\hfill & \\ & =\upsilon (r+1)-2r-2+2+2r+2\hfill & \\ & =\upsilon (r+1)+2.\hfill \end{array}$$

Also, by induction on $\upsilon$, we have

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}^{**}/(J^{*},x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -1,r}^{*}/I_{\upsilon -1,r}^{*})+2r+1\hfill & \\ & =(\upsilon -1)(r+1)+1+2r+1\hfill & \\ & =\upsilon (r+1)-r-1+1+2r+1\hfill & \\ & =\upsilon (r+1)+r+1.\hfill \end{array}$$

By the Depth Lemma, $depth({\mathsf{F}}_{\upsilon ,r}^{**}/J^{*})\ge \upsilon (r+1)+2.$ It is easy to see that ${\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},y_{\upsilon +2})$$\cong {\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*}{⨂}_{K}K[y_{(\upsilon +2)1},y_{(\upsilon +2)2},\dots ,y_{(\upsilon +2)r}].$ By Lemma 3, we have $depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r},$$y_{\upsilon +2}\left)\right)=depth({\mathsf{F}}_{\upsilon ,r}^{*}/I_{\upsilon ,r}^{*})+r=\upsilon (r+1)+1+r$. Using the Depth Lemma, $depth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})$$\ge \upsilon (r+1)+2$. For upper bound as $x_{\upsilon }\notin E_{\upsilon ,r}$, and

$${\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:x_{\upsilon })\cong {\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon -2,r}\underset{K}{⨂}K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right)\underset{K}{⨂}K[x_{\upsilon },y_{(\upsilon +1)1},\dots ,y_{(\upsilon +1)r},x_{(\upsilon -1)1},\dots ,x_{(\upsilon -1)r}].$$

Thus, by Lemmas 3 and 4 and induction on $\upsilon$,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon -2,r})+depth(K\left[V\left({\mathcal{S}}_{r+1}\right)\right]/I\left({\mathcal{S}}_{r+1}\right))+2r+1\hfill & \\ & =(\upsilon -2)(r+1)+2+1+2r+1\hfill & \\ & =\upsilon (r+1)-2r-2+2+2r+2\hfill & \\ & =\upsilon (r+1)+2.\hfill \end{array}$$

Using Corollary 1, $depth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})\le depth({\mathsf{F}}_{\upsilon ,r}^{**}/(E_{\upsilon ,r}:x_{\upsilon }))=\upsilon (r+1)+2.$ This completes the proof for depth.

For Stanley depth, when $\upsilon =1$, by applying Lemma 2 instead of the Depth Lemma and Lemma 6 instead of Lemma 5 on the short exact sequence, we obtain $sdepth({\mathsf{F}}_{1,r}^{**}/E_{1,r})\ge r+3.$ For upper bound, consider $\mu =y_{21}\dots y_{2r}{y}_1{y}_3{x}_1\in {\mathsf{F}}_{1,r}^{**}/E_{1,r}$; clearly $x\mu \in E_{1,r}$, for all $x\in supp\left(E_{1,r}\right)\setminus supp\left(\mu \right)$. Therefore, by Lemma 7, $sdepth({\mathsf{F}}_{1,r}^{**}/E_{1,r})\le r+3$. For $\upsilon \ge 2$, the required result follows by applying Lemma 2 instead of the Depth Lemma, Lemma 6 instead of Lemma 5, and Proposition 1 instead of Corollary 1. If $\upsilon$ is even, then we obtain $sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})\ge (\upsilon +1)(r+1)$. For upper bound, consider

$$\begin{array}{cc}\hfill \mu =y_{11}\dots & y_{1r}\dots y_{(\upsilon -1)1}\dots y_{(\upsilon -1)r}{y}_{(\upsilon +1)1}\dots y_{(\upsilon +1)r}{x}_{11}\dots x_{1r}\dots \hfill \\ & x_{(\upsilon -3)1}\dots x_{(\upsilon -3)r}{x}_{(\upsilon -1)1}\dots x_{(\upsilon -1)r}{y}_2{y}_4\dots y_{\upsilon }{y}_{\upsilon +2}{x}_2{x}_4\dots x_{\upsilon -2}{x}_{\upsilon }\in {\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r}.\hfill \end{array}$$

Clearly $x\mu \in E_{\upsilon ,r}$, for all $x\in supp\left(E_{\upsilon ,r}\right)\setminus supp\left(\mu \right)$; therefore, by Lemma 7, $sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})$$\le (\upsilon +1)r+\upsilon +1=(\upsilon +1)(r+1)$. Hence, $sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})=(\upsilon +1)(r+1).$ If $\upsilon$ is odd, then we obtain $sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})\ge \upsilon (r+1)+2$. For upper bound, consider

$$\begin{array}{cc}\hfill \mu =y_{21}\dots & y_{2r}\dots y_{(\upsilon -1)1}\dots y_{(\upsilon -1)r}{y}_{(\upsilon +1)1}\dots y_{(\upsilon +1)r}{x}_{21}\dots x_{2r}\dots \hfill \\ & x_{(\upsilon -3)1}\dots x_{(\upsilon -3)r}{x}_{(\upsilon -1)1}\dots x_{(\upsilon -1)r}{y}_1{y}_3\dots y_{\upsilon }{y}_{\upsilon +2}{x}_1{x}_3\dots x_{\upsilon -2}{x}_{\upsilon }\in {\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r}.\hfill \end{array}$$

Clearly $x\mu \in E_{\upsilon ,r}$, for all $x\in supp\left(E_{\upsilon ,r}\right)\setminus supp\left(\mu \right)$; therefore, by Lemma 7, $sdepth({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})$$\le \upsilon r+\upsilon +2=\upsilon (r+1)+2$. This completes the proof for Stanley depth. □

Corollary 5.

Let $r\ge 1$ and $\upsilon \ge 1$. Then

$$pdim({\mathsf{F}}_{\upsilon ,r}^{**}/E_{\upsilon ,r})=\left\{\begin{array}{cc}(\upsilon +1)(r+1),\hfill & \mathit{if}\upsilon \mathit{is}\mathit{even};\hfill \\ \upsilon (r+1)+2r,\hfill & \mathit{if}\upsilon \mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

Proof. 

The required result can be obtained by using Theorem 5 and Theorem 1. □

Now we find depth, Stanley depth, and projective dimension of the edge ideals of the r-fold bristled graph of the circular ladder graph.

Theorem 6.

Let $\upsilon \ge 3$ and $r\ge 1$. Then

$$depth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=\left\{\begin{array}{cc}\upsilon (r+1),\hfill & \mathit{if}\upsilon \mathit{is}\mathit{even};\hfill \\ \upsilon (r+1)+r-1,\hfill & \mathit{if}\upsilon \mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

Proof. 

First we will prove the result for depth. Consider the following short exact sequence:

$$\begin{array}{c}\hfill 0⟶{\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon })\stackrel{·x_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r}⟶{\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r},x_{\upsilon })⟶0.\end{array}$$

(5)

After a suitable renumbering of variables, we have

$${\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r}\underset{K}{⨂}K[\left\{x_{\upsilon }\right\}\bigcup \bigcup _{j=1}^r\{x_{(\upsilon -2)j},x_{(\upsilon -1)j},y_{\upsilon j}\}].$$

By Lemma 3,

$$depth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1.$$

Let $A^{*}:=({\mathfrak{C}}_{\upsilon ,r},x_{\upsilon })$ and $\mathcal{G}\left(A^{*}\right)=\mathcal{G}\left(I_{\upsilon -1,r}\right)\bigcup \{y_1{y}_{\upsilon },y_{\upsilon }{y}_{\upsilon -1},x_{\upsilon }\}\bigcup \{y_{\upsilon }{y}_{\upsilon 1},y_{\upsilon }{y}_{\upsilon 2},\dots ,$$y_{\upsilon }{y}_{\upsilon r}\}.$ Consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon })\stackrel{·y_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/A^{*}⟶{\mathsf{F}}_{\upsilon ,r}/(A^{*},y_{\upsilon })⟶0.$$

After renumbering of variables, we have

$${\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r}\underset{K}{⨂}K[\left\{y_{\upsilon }\right\}\bigcup \bigcup _{j=1}^r\{x_{\upsilon j},y_{(\upsilon -2)j},y_{(\upsilon -1)j}\}],$$

and

$${\mathsf{F}}_{\upsilon ,r}/(A^{*},y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}/I_{\upsilon -1,r}\underset{K}{⨂}K[x_{\upsilon 1},x_{\upsilon 2},\dots ,x_{\upsilon r},y_{\upsilon 1},y_{\upsilon 2},\dots ,y_{\upsilon r}].$$

Case 1. When $\upsilon$ is even, using Lemma 3, $depth({\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1$. As $\upsilon$ is even, so $\upsilon -3$ will be an odd number. So by Theorem 5, we have

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon }))& =(\upsilon -3)(r+1)+2+3r+1\hfill & \\ & =\upsilon (r+1)-3r-3+3r+3\hfill & \\ & =\upsilon (r+1).\hfill \end{array}$$

Similarly, by Lemma 3 and Theorem 3,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(A^{*},y_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -1,r}/I_{\upsilon -1,r})+2r\hfill & \\ & =(\upsilon -1)(r+1)+2r\hfill & \\ & =\upsilon (r+1)-r-1+2r\hfill & \\ & =\upsilon (r+1)+r-1.\hfill \end{array}$$

By the Depth Lemma, $depth({\mathsf{F}}_{\upsilon ,r}/A^{*})\ge \upsilon (r+1).$ Now by Theorem 5,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1\hfill & \\ & =(\upsilon -3)(r+1)+2+3r+1\hfill & \\ & =\upsilon (r+1)-3r-3+3r+3\hfill & \\ & =\upsilon (r+1).\hfill \end{array}$$

Applying the Depth Lemma on short exact sequence 5, we obtain $depth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=\upsilon (r+1).$ This completes the proof when $\upsilon$ is even.

Case 2. If $\upsilon$ is odd, using Lemma 3, $depth({\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1$. As $\upsilon$ is odd, so $\upsilon -3$ will be an even number. So by Theorem 5, we have

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(A^{*}:y_{\upsilon }))& =(n-3+1)(r+1)+3r+1\hfill & \\ & =\upsilon (r+1)-2r-2+3r+1\hfill & \\ & =\upsilon (r+1)+r-1.\hfill \end{array}$$

Now by Lemma 3 and Theorem 3,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(A^{*},y_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -1,r}/I_{\upsilon -1,r})+2r\hfill & \\ & =(\upsilon -1)(r+1)+2r\hfill & \\ & =\upsilon (r+1)-r-1+2r\hfill & \\ & =\upsilon (r+1)+r-1.\hfill \end{array}$$

By the Depth Lemma, $depth({\mathsf{F}}_{\upsilon ,r}/A^{*})=\upsilon (r+1)+r-1.$ By Theorem 5,

$$\begin{array}{ccc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1\hfill & \\ & =(\upsilon -3+1)(r+1)+3r+1\hfill & \\ & =\upsilon (r+1)-2r-2+3r+1\hfill & \\ & =\upsilon (r+1)+r-1.\hfill \end{array}$$

Applying the Depth Lemma on short exact sequence 5, we obtain $depth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=\upsilon (r+1)+r-1.$ This completes the proof for depth.

For Stanley depth, the required result follows by applying Lemma 2 instead of the Depth Lemma and Lemma 6 instead of Lemma 5. When $\upsilon$ is even, we have $sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})$$\ge \upsilon (r+1)$. For upper bound as $x_{\upsilon }\notin {\mathfrak{C}}_{\upsilon ,r}$ and $sdepth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))=$$sdepth({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1$, by Theorem 5 and Proposition 1 $sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})\le sdepth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))=\upsilon (r+1).$ Similarly, when $\upsilon$ is odd, we obtain $sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})$$\ge \upsilon (r+1)+r-1.$ For upper bound as $x_{\upsilon }\notin {\mathfrak{C}}_{\upsilon ,r}$ and $sdepth({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))=sdepth$$({\mathsf{F}}_{\upsilon -3,r}^{**}/E_{\upsilon -3,r})+3r+1$, by Theorem 5 and Proposition 1, $sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})\le sdepth$$({\mathsf{F}}_{\upsilon ,r}/({\mathfrak{C}}_{\upsilon ,r}:x_{\upsilon }))=\upsilon (r+1)+r-1$. Hence,

$$sdepth({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=\upsilon (r+1)+r-1.$$

□

Corollary 6.

Let $\upsilon \ge 3$ and $r\ge 1$. Then

$$pdim({\mathsf{F}}_{\upsilon ,r}/{\mathfrak{C}}_{\upsilon ,r})=\left\{\begin{array}{cc}\upsilon (r+1),\hfill & \mathit{if}\upsilon \mathit{is}\mathit{even};\hfill \\ \upsilon (r+1)-r+1,\hfill & \mathit{if}\upsilon \mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

Proof. 

The required result can be obtain by using Theorem 6 and Theorem 1. □

We also have formulae for values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals of the $T_{\upsilon }$ graph, as given in the next theorem and corollary.

Theorem 7.

Let $\upsilon \ge 3$ and $r\ge 1$. Then

$$depth({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})=sdepth({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$$

Proof. 

First we will prove the result for depth. We will prove this for $\upsilon \ge 3$. Consider the following short exact sequence:

$$\begin{array}{c}\hfill 0⟶{\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r}:x_{\upsilon })\stackrel{·x_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r}⟶{\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r},x_{\upsilon })⟶0.\end{array}$$

(6)

After renumbering the variables, we have

$${\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r}:x_{\upsilon })\cong {\mathsf{F}}_{\upsilon -3,r}/L_{\upsilon -3,r}\underset{K}{⨂}K[\left\{x_{\upsilon }\right\}\bigcup \bigcup _{j=1}^r\{x_{(\upsilon -2)j},x_{(\upsilon -1)j},y_{(\upsilon -2)j},y_{\upsilon j},y_{(\upsilon -1)j}\}].$$

Using Lemma 3 and Theorem 4,

$$\begin{array}{cc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r}:x_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -3,r}/L_{\upsilon -3,r})+5r+1\hfill \\ \hfill & =\lfloor \frac{3(\upsilon -3)}{2}\rfloor r+\lceil \frac{\upsilon -3}{2}\rceil +5r+1\hfill \\ \hfill & =\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .\hfill \end{array}$$

Let $J^{\prime }:=(C_{\upsilon ,r},x_{\upsilon })$, where $\mathcal{G}\left(J^{\prime }\right)=\mathcal{G}\left(I_{\upsilon -1}\right)\bigcup \{x_{\upsilon -1}{y}_{\upsilon },y_{\upsilon -1}{y}_{\upsilon },y_{\upsilon }{y}_1,y_{\upsilon }{x}_1,x_{\upsilon }\}{\bigcup }_{j=1}^r$$\left\{y_{\upsilon }{y}_{\upsilon j}\right\}.$ Consider the following short exact sequence:

$$0⟶{\mathsf{F}}_{\upsilon ,r}/(J^{\prime }:y_{\upsilon })\stackrel{·y_{\upsilon }}{\to }{\mathsf{F}}_{\upsilon ,r}/J^{\prime }⟶{\mathsf{F}}_{\upsilon ,r}/(J^{\prime },y_{\upsilon })⟶0.$$

After renumbering the variables, we have

$$({\mathsf{F}}_{\upsilon ,r}/(J^{\prime }:y_{\upsilon }))\cong {\mathsf{F}}_{\upsilon -3,r}/L_{\upsilon -3,r}\underset{K}{⨂}K[\left\{y_{\upsilon }\right\}\bigcup \bigcup _{j=1}^r\{y_{(\upsilon -2)j},x_{(\upsilon -2)j},x_{\upsilon j},x_{(\upsilon -1)j},y_{(\upsilon -1)j}\}].$$

By Lemma 3 and Theorem 4,

$$\begin{array}{cc}\hfill depth({\mathsf{F}}_{\upsilon ,r}/(J^{\prime }:y_{\upsilon }))& =depth({\mathsf{F}}_{\upsilon -3,r}/L_{\upsilon -3,r})+5r+1\hfill \\ \hfill & =\lfloor \frac{3(\upsilon -3)}{2}\rfloor r+\lceil \frac{\upsilon -3}{2}\rceil +5r+1\hfill \\ \hfill & =\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .\hfill \end{array}$$

Now $\mathcal{G}(J^{\prime },y_{\upsilon })=\mathcal{G}\left(L_{\upsilon -1,r}\right)\bigcup \{y_{\upsilon },x_{\upsilon }\}$ and ${\mathsf{F}}_{\upsilon ,r}/(J^{\prime },y_{\upsilon })\cong {\mathsf{F}}_{\upsilon -1,r}/L_{\upsilon -1,r}{⨂}_{K}K[{\bigcup }_{j=1}^r$$\{x_{\upsilon j},$$y_{\upsilon j}\left\}\right].$ Using Lemma 3 and Theorem 4, we have $depth({\mathsf{F}}_{\upsilon ,r}/(J^{\prime },y_{\upsilon }))=depth({\mathsf{F}}_{\upsilon -1,r}/$$L_{\upsilon -1,r})+2r=\lfloor \frac{3(\upsilon -1)}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil +2r.=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$ By the Depth Lemma,

$$depth({\mathsf{F}}_{\upsilon ,r}/J^{\prime })=depth({\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r},x_{\upsilon }))=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$$

Applying the Depth Lemma on short exact sequence 6, $depth({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$ This completes the proof for depth.

For Stanley depth, the required result follows by applying Lemma 2 instead of the Depth Lemma. We obtain $sdepth({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})\ge \lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$ For upper bound as $x_{\upsilon }\notin C_{\upsilon ,r}$ we have

$$sdepth({\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r}:x_{\upsilon }))=sdepth({\mathsf{F}}_{\upsilon -3,r}/L_{\upsilon -3,r})+5r+1,$$

by Theorem 4 and Proposition 1,

$$sdepth({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})\le sdepth({\mathsf{F}}_{\upsilon ,r}/(C_{\upsilon ,r}:x_{\upsilon }))=\lfloor \frac{3\upsilon +1}{2}\rfloor r+\lceil \frac{\upsilon -1}{2}\rceil .$$

This completes the proof. □

Corollary 7.

Let $\upsilon \ge 3$ and $r\ge 1$. Then

$$pdim({\mathsf{F}}_{\upsilon ,r}/C_{\upsilon ,r})=\lceil \frac{\upsilon -1}{2}\rceil r+\lfloor \frac{3\upsilon +1}{2}\rfloor .$$

Proof. 

The required result can be obtain by using Theorem 7 and Theorem 1. □

Example 2.

If $\upsilon =9$ and $r=4$, then by Theorem 7, we have $depth({\mathsf{F}}_{9,4}/C_{9,4})=sdepth$$({\mathsf{F}}_{9,4}/C_{9,4})$$=\lfloor \frac{3\left(9\right)+1}{2}\rfloor \left(4\right)+\lceil \frac{9-1}{2}\rceil =56+4=60$. Also, by Corollary 4 we have pdim $({\mathsf{F}}_{9,4}/C_{9,4})=\lceil \frac{9-1}{2}\rceil \left(4\right)+\lfloor \frac{3\left(9\right)+1}{2}\rfloor =16+14=30.$