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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 23593

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Special Issue Editors

Prof. Dr. Weifan Wang

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Guest Editor

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Interests: graph coloring; graph labeling; graph partition; surviving rate; connectivity
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Min Chen

E-Mail Website
Guest Editor

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Interests: graph coloring; arboricity; forest partition; planar graph; graph embedding
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Yongtang Shi

E-Mail Website
Guest Editor

Center for Combinatorics, Nankai University, Tianjin 300071, China
Interests: graph theory and its applications; combinatorial optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Graph and hypergraph theory is one of the most rapidly evolving fields of theoretical aspects of the study of discrete structures, and its applications are widely expanded in various areas, including computer science, artificial intelligence, data science, statistical physics, and chemistry. Symmetry is a basic attribute of aesthetic appreciation. A number of different symmetric measurements for networks and graphs have been developed and analyzed, becoming an important criterion that illustrates the structure and properties of graphs. The differences are due in part to the fact that symmetry can be interpreted in different ways, e.g., by means of knot theory or the automorphism group of a graph. Recently, symmetric measurements have been applied in many disciplines. Based on vertex orbits, it has long been used to define measures of the structural complexity of graphs and hypergraphs. Algebraic graph theory is a classical field where symmetry has been investigated extensively and the role of symmetry in network aesthetics attracts much more attention. In this Special Issue, we would like to invite you to submit your original research on the theory and applications of symmetry in graph and hypergraph theory.

Topics of interest include but are not limited to the following:

Graph and hypergraph;
Networks;
Coloring and labeling;
Partition and cover;
Ramsey theory;
Caylay graph and symmetric graph;
Extreme value problems;
Topological indices;
Graph algorithms;
Algebraic tools for graphs and hypergraphs;

Prof. Dr. Weifan Wang
Prof. Dr. Min Chen
Prof. Dr. Yongtang Shi
Guest Editors

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Published Papers (14 papers)

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18 pages, 21610 KiB

Open AccessArticle

Extended Graph of Fuzzy Topographic Topological Mapping Model:

G_{0}^{4} ({F T T M}_{n}^{4})

by Noorsufia Abd Shukor, Tahir Ahmad, Amidora Idris, Siti Rahmah Awang, Muhammad Zillullah Mukaram and Norma Alias

Symmetry 2022, 14(12), 2645; https://doi.org/10.3390/sym14122645 - 15 Dec 2022

Cited by 1 | Viewed by 1338

Abstract

Fuzzy topological topographic mapping (

F T T M

) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate [...] Read more.

Fuzzy topological topographic mapping (

F T T M

F T T M_{n}

, is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. The special characteristic of

F T T M

, namely, the homeomorphisms between its components, allows the generation of new

F T T M

. The generated

F T T M

s can be represented as pseudo graphs. A pseudo-graph consists of vertices that signify the generated

F T T M

and edges that connect their incidence components. A graph of pseudo degree zero,

G_{0} (F T T M_{n}^{k}),

however, is a special type of graph where each of the

F T T M

components differs from its adjacent. A researcher posted a conjecture on

G_{0}^{3} (F T T M_{n}^{3})

in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended

G_{0}^{3} (F T T M_{n}^{3})

, namely, the conjecture on

G_{0}^{4} (F T T M_{n}^{4})

that was posed in 2018, is narrated and proven using simple mathematical induction. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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14 pages, 821 KiB

Open AccessArticle

A Further Study on the Degree-Corrected Spectral Clustering under Spectral Graph Theory

by Fangmeng Liu, Wei Li and Yiwen Zhong

Symmetry 2022, 14(11), 2428; https://doi.org/10.3390/sym14112428 - 16 Nov 2022

Viewed by 1771

Abstract

Spectral clustering algorithms are often used to find clusters in the community detection problem. Recently, a degree-corrected spectral clustering algorithm was proposed. However, it is only used for partitioning graphs which are generated from stochastic blockmodels. This paper studies the degree-corrected spectral clustering algorithm based on the spectral graph theory and shows that it gives a good approximation of the optimal clustering for a wide class of graphs. Moreover, we also give theoretical support for finding an appropriate degree-correction. Several numerical experiments for community detection are conducted in this paper to evaluate our method. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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7 pages, 266 KiB

Open AccessArticle

Online and Connected Online Ramsey Numbers of a Matching versus a Path

by Ruyu Song and Yanbo Zhang

Symmetry 2022, 14(11), 2277; https://doi.org/10.3390/sym14112277 - 31 Oct 2022

Viewed by 1204

Abstract

The

(G_{1}, G_{2})

-online Ramsey game is a two-player turn-based game between a builder and a painter. Starting from an empty graph with infinite vertices, the builder adds a new edge in each round, and the painter colors [...] Read more.

The

(G_{1}, G_{2})

G_{1}

or a blue copy of

G_{2}

in as few rounds as possible, while the painter’s aim is the opposite. The online Ramsey number

\tilde{r} (G_{1}, G_{2})

is the minimum number of edges that the builder needs to win the

(G_{1}, G_{2})

-online Ramsey game, regardless of the painter’s strategy. Furthermore, we initiate the study of connected online Ramsey game, which is identical to the usual one, except that at any time the graph induced by all edges should be connected. In this paper, we show a general bound of the online Ramsey number of a matching versus a path and determine its exact value when the path has an order of three or four. For the connected version, we obtain all connected online Ramsey numbers of a matching versus a path. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

13 pages, 360 KiB

Open AccessArticle

The l₁-Embeddability of Hypertrees and Unicyclic Hypergraphs

by Guangfu Wang, Lijun Chen and Zhikun Xiong

Symmetry 2022, 14(11), 2260; https://doi.org/10.3390/sym14112260 - 27 Oct 2022

Cited by 2 | Viewed by 1379

Abstract

A hypercube is a graph whose nodes can be labeled by binary vectors such that the distance between the binary addresses in the graph is the Hamming distance. Due to the symmetry of the hypercube, one usually considers the graph embedded in the [...] Read more.

l_{1}

-graphs. In this paper, we determine the

l_{1}

-embeddability of hypertrees and unicyclic hypergraphs. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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8 pages, 325 KiB

Open AccessArticle

The g-Extra Connectivity of the Strong Product of Paths and Cycles

by Qinze Zhu and Yingzhi Tian

Symmetry 2022, 14(9), 1900; https://doi.org/10.3390/sym14091900 - 11 Sep 2022

Cited by 4 | Viewed by 1343

Abstract

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if

G - S

is disconnected and each component of

G - S

has at least [...] Read more.

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if

G - S

is disconnected and each component of

G - S

has at least

g + 1

vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs

G_{1} = (V_{1}, E_{1})

and

G_{2} = (V_{2}, E_{2})

, the strong product

G_{1} ⊠ G_{2}

is defined as follows: its vertex set is

V_{1} \times V_{2}

and its edge set is

{(x_{1}, x_{2}) (y_{1}, y_{2}) |

x_{1} = x_{2}

and

y_{1} y_{2} \in E_{2}

; or

y_{1} = y_{2}

and

x_{1} x_{2} \in E_{1}

; or

x_{1} x_{2} \in E_{1}

and

y_{1} y_{2} \in E_{2}}

, where

(x_{1}, x_{2}), (y_{1}, y_{2}) \in V_{1} \times V_{2}

. In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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16 pages, 420 KiB

Open AccessArticle

Extremal Structure on Revised Edge-Szeged Index with Respect to Tricyclic Graphs

by Tongkun Qu and Shengjin Ji

Symmetry 2022, 14(8), 1646; https://doi.org/10.3390/sym14081646 - 10 Aug 2022

Viewed by 1358

Abstract

For a given graph G, [...] Read more.

For a given graph G,

S z_{e}^{*} (G) = \sum_{e = u v \in E (G)} (m_{u} (e) + \frac{m_{0} (e)}{2}) (m_{v} (e) + \frac{m_{0} (e)}{2})

is the revised edge-Szeged index of G, where

m_{u} (e)

and

m_{v} (e)

are the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, respectively, and

m_{0} (e)

is the number of edges equidistant to u and v. In this paper, we identify the lower bound of the revised edge-Szeged index among all tricyclic graphs and also characterize the extremal structure of graphs that attain the bound. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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16 pages, 337 KiB

Open AccessArticle

Graph Coloring via Clique Search with Symmetry Breaking

by Sándor Szabó and Bogdán Zaválnij

Symmetry 2022, 14(8), 1574; https://doi.org/10.3390/sym14081574 - 30 Jul 2022

Cited by 3 | Viewed by 1584

Abstract

It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the symmetries present in the auxiliary graph. The proposed method can also be used for efficient exact coloring of hyper graphs. We also precondition the auxiliary graph in order to further reduce the search space. We carry out numerical experiments to assess the practicality of these proposals. We solve some hard cases and prove a new lower limit of seven for the mycielski7 graph with the aid of the proposed technique. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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9 pages, 321 KiB

Open AccessArticle

A Note on the Lagrangian of Linear 3-Uniform Hypergraphs

by Sinan Hu and Biao Wu

Symmetry 2022, 14(7), 1402; https://doi.org/10.3390/sym14071402 - 7 Jul 2022

Cited by 2 | Viewed by 1532

Abstract

Lots of symmetric properties are well-explored and analyzed in extremal graph theory, such as the well-known symmetrization operation in the Turán problem and the high symmetric in the extremal graphs. This paper is devoted to studying the Lagrangian of hypergraphs, which connects to [...] Read more.

π_{λ} (F)

is the limit supremum of

r! λ (G)

over all F-free G, where

λ (G)

is the Lagrangian of G. An r-uniform hypergraph F is called

λ

-perfect if

π_{λ} (F)

equals

r! λ (K_{v (F) - 1}^{r})

. Yan and Peng conjectured that: for integer

r \geq 3

, there exists

n_{0} (r)

such that if G and H are two

λ

-perfect r-graphs with

| V (G) |

and

| V (H) |

no less than

n_{0} (r)

, then the disjoint union of G and H is

λ

-perfect. Let

S_{t}

denote a 3-uniform hypergraph with t edges

{e_{1}, \dots, e_{t}}

satisfying that

e_{i} \cap e_{j} = {v}

for all

1 \leq i < j \leq t

. In this paper, we show that the conjecture holds for

G = S_{2}

and

H = S_{t}

for all

t \geq 62

. Moreover, our result yields a class of Turán densities of 3-uniform hypergraphs. In our proof, we use some new techniques to study Lagrangian density problems; using a result of Sidorenko to find subgraphs, and a result of Talbot to upper bound the Lagrangian of a hypergraph. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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6 pages, 275 KiB

Open AccessArticle

Some Results on the Erdős–Faber–Lovász Conjecture

by Yun Feng and Wensong Lin

Symmetry 2022, 14(7), 1327; https://doi.org/10.3390/sym14071327 - 27 Jun 2022

Viewed by 1697

Abstract

Erdős–Faber–Lovász conjecture states that if a graph G is a union of the n edge-disjoint copies of complete graph

K_{n}

, that is, each pair of complete graphs has at most one shared vertex, then the chromatic number of graph G is [...] Read more.

Erdős–Faber–Lovász conjecture states that if a graph G is a union of the n edge-disjoint copies of complete graph

K_{n}

, that is, each pair of complete graphs has at most one shared vertex, then the chromatic number of graph G is n. In fact, we only need to consider the graphs where each pair of complete graphs has exactly one shared vertex. However, each shared vertex may be shared by more than two complete graphs. Therefore, this paper first considers the graphs where each shared vertex happens to be shared by two complete graphs, and then discusses the graphs with only one shared vertex shared by more than two complete graphs. The conjecture is correct for these two kinds of graphs in this work. Finally, the graph where each shared vertex happens to be shared by three complete graphs has been studied, and the conjecture also holds for such graphs when

n = 13

. The graphs discussed in this paper have certain symmetric properties. The symmetry of graphs plays an important role in coloring. This work is an attempt to combine the symmetry of graphs with the coloring of graphs. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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17 pages, 857 KiB

Open AccessArticle

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

by Jingjing Huo, Mingchao Li and Ying Wang

Symmetry 2022, 14(7), 1289; https://doi.org/10.3390/sym14071289 - 21 Jun 2022

Cited by 2 | Viewed by 1261

Abstract

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index

χ_{a}^{'} (G)

of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and

G \neq C_{5}

, then

χ_{a}^{'} (G) \leq Δ + 2

. In this paper, we show that if G is a planar graph with maximum degree

Δ \geq 13

, then

Δ \leq χ_{a}^{'} (G) \leq Δ + 1

, and, further,

χ_{a}^{'} (G) = Δ + 1

if and only if G contains two adjacent vertices of maximum degree. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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9 pages, 386 KiB

Open AccessArticle

The Surviving Rate of IC-Planar Graphs

by Xiaoxue Hu, Jiacheng Hu and Jiangxu Kong

Symmetry 2022, 14(6), 1258; https://doi.org/10.3390/sym14061258 - 17 Jun 2022

Viewed by 1509

Abstract

Let k and n be two positive integers. Firefighting is a discrete dynamical process of preventing the spread of fire. Let G be a connected graph G with n vertices. Assuming a fire starts at one of the vertices of G, the firefighters choose k unburned vertices at each step, and then the fire spreads to all unprotected neighbors of the burning vertices. The process continues until the fire stops spreading. The goal is to protect as many vertices as possible. When a fire breaks out randomly at a vertex of G, its k-surviving rate,

ρ_{k} (G)

, is the expected number of saved vertices. A graph is IC-planar if it has a drawing in which each edge cross once and their endpoints are disjoint. In this paper, we prove that

ρ_{4} (G) > \frac{1}{124}

for every IC-planar graph G. This is proven by the discharging method and the locally symmetric of the graph. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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14 pages, 337 KiB

Open AccessArticle

The Outer-Planar Anti-Ramsey Number of Matchings

by Changyuan Xiang, Yongxin Lan, Qinghua Yan and Changqing Xu

Symmetry 2022, 14(6), 1252; https://doi.org/10.3390/sym14061252 - 16 Jun 2022

Viewed by 1600

Abstract

A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let

a r (G, H)

denote the maximum positive integer t, such that there is a t-edge-colored graph G [...] Read more.

A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let

a r (G, H)

denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by

k K_{2}

a matching of size k and

O_{n}

the class of all maximal outer-planar graphs on n vertices, respectively. The outer-planar anti-Ramsey number of graph H, denoted by

a r (O_{n}, H)

, is defined as

max {a r (O_{n}, H) | O_{n} \in O_{n}}

. It seems nontrivial to determine the exact values for

a r (O_{n}, H)

because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that

a r (O_{n}, k K_{2}) \leq n + 3 k - 8

for all

n \geq 2 k

and

k \geq 6

, which improves the existing upper bound for

a r (O_{n}, k K_{2})

, and prove that

a r (O_{n}, k K_{2}) = n + 2 k - 5

for

n = 2 k

and

k \geq 5

. We also obtain that

a r (O_{n}, 6 K_{2}) = n + 6

for all

n \geq 29

. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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11 pages, 380 KiB

Open AccessArticle

Star Chromatic Index of 1-Planar Graphs

by Yiqiao Wang, Juan Liu, Yongtang Shi and Weifan Wang

Symmetry 2022, 14(6), 1177; https://doi.org/10.3390/sym14061177 - 8 Jun 2022

Cited by 1 | Viewed by 1792

Abstract

Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of specific colorings of a graph. This [...] Read more.

χ_{st}^{'} (G)

of a graph G is defined as the smallest k for which the edges of G can be colored by using k colors so that no two adjacent edges get the same color and no bichromatic paths or cycles of length four are produced. A graph G is called 1-planar if it can be drawn in the plane such that each edge crosses at most one other edge. In this paper, we prove that every 1-planar graph G satisfies

χ_{st}^{'} (G) \leq 7.75 Δ + 166

; and moreover

χ_{st}^{'} (G) \leq ⌊ 1.5 Δ ⌋ + 500

if G contains no 4-cycles, and

χ_{st}^{'} (G) \leq 2.75 Δ + 116

if G is 3-connected, or optimal, or NIC-planar. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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Review

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15 pages, 415 KiB

Open AccessReview

The Genus of a Graph: A Survey

by Liangxia Wan

Symmetry 2023, 15(2), 322; https://doi.org/10.3390/sym15020322 - 23 Jan 2023

Viewed by 2119

Abstract

The problem of determining the genus for a graph can be dated to the Map Color Conjecture proposed by Heawood in 1890. This was implied to be a Thread Problem by Hilbert and Cohn-Vossen. The conjecture was finally established by Ringel, Youngs, and many other mathematicians. Subsequently, the genera of some special graphs with symmetry were determined. The study of genus embeddings of graphs is closely related to other invariants of a graph. Specifically, the computational complexity is dependent on the genus of the underlying graph for certain well-known NP-hard problems. In this survey, main construction techniques and certain criteria are stated in the topic of the genus of a graph. Most graphs with a known genus are listed. A new theorem is shown that the method of joint trees of a graph is reasonable. Moreover, a formal set is introduced, and related results are obtained. Although a cubic graph of Hamilton cycle is asymmetric, it is interesting that a set of associate surfaces of all its joint trees is a formal set with symmetry. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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