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Symmetry
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Advances in Graph Theory
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Advances in Graph Theory

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

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 5332

Special Issue Editor

Dr. Yuefang Sun

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
Interests: graph theory; combinatorial optimization; algorithms and complexity analysis

Special Issue Information

Dear Colleagues,

Graph theory is now one of the most active branches of mathematics. In recent decades, much progress has been made in both the theoretical research and practical applications of graph theory.

The goal of this Special Issue is to collect original research articles on this subject. Submissions related to all aspects of graph theory that present new results, especially on symmetric phenomena, are welcome.

The topics of interest include, but are not limited to, the following:

  • Structural graph theory;
  • Extremal graph theory;
  • Random graph theory;
  • Spectral and algebraic graph theory;
  • Chemical graph theory;
  • Topological graph theory;
  • Graph limits theory;
  • Graph algorithms and complexity analysis.

Dr. Yuefang Sun
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

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Research

17 pages, 322 KiB  
Article
BDAC: Boundary-Driven Approximations of K-Cliques
by Büşra Çalmaz and Belgin Ergenç Bostanoğlu
Symmetry 2024, 16(8), 983; https://doi.org/10.3390/sym16080983 - 2 Aug 2024
Viewed by 958
Abstract
Clique counts are crucial in applications like detecting communities in social networks and recurring patterns in bioinformatics. Counting k-cliques—a fully connected subgraph with k nodes, where each node has a direct, mutual, and symmetric relationship with every other node—becomes computationally challenging for larger [...] Read more.
Clique counts are crucial in applications like detecting communities in social networks and recurring patterns in bioinformatics. Counting k-cliques—a fully connected subgraph with k nodes, where each node has a direct, mutual, and symmetric relationship with every other node—becomes computationally challenging for larger k due to combinatorial explosion, especially in large, dense graphs. Existing exact methods have difficulties beyond k = 10, especially on large datasets, while sampling-based approaches often involve trade-offs in terms of accuracy, resource utilization, and efficiency. This difficulty becomes more pronounced in dense graphs as the number of potential k-cliques grows exponentially. We present Boundary-driven approximations of k-cliques (BDAC), a novel algorithm that approximates k-clique counts without using recursive procedures or sampling methods. BDAC offers both lower and upper bounds for k-cliques at local (per-vertex) and global levels, making it ideal for large, dense graphs. Unlike other approaches, BDAC’s complexity remains unaffected by the value of k. We demonstrate its effectiveness by comparing it with leading algorithms across various datasets, focusing on k values ranging from 8 to 50. Full article
(This article belongs to the Special Issue Advances in Graph Theory)
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15 pages, 301 KiB  
Article
Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs
by Xiaosha Wei
Symmetry 2024, 16(4), 497; https://doi.org/10.3390/sym16040497 - 19 Apr 2024
Viewed by 670
Abstract
Let D=(V(D),A(D)) be a digraph of order n and let r∈S⊆V(D) with 2≤|S|≤n. A directed [...] Read more.
Let D=(V(D),A(D)) be a digraph of order n and let r∈S⊆V(D) with 2≤|S|≤n. A directed (S,r)-Steiner path (or an (S,r)-path for short) is a directed path P beginning at r such that S⊆V(P). Arc-disjoint between two (S,r)-paths is characterized by the absence of common arcs. Let λS,rp(D) be the maximum number of arc-disjoint (S,r)-paths in D. The directed path k-arc-connectivity of D is defined as λkp(D)=min{λS,rp(D)∣S⊆V(D),S=k,r∈S}. In this paper, we shall investigate the directed path 3-arc-connectivity of Cartesian product λ3p(G□H) and prove that if G and H are two digraphs such that δ0(G)≥4, δ0(H)≥4, and κ(G)≥2, κ(H)≥2, then λ3p(G□H)≥min2κ(G),2κ(H); moreover, this bound is sharp. We also obtain exact values for λ3p(G□H) for some digraph classes G and H, and most of these digraphs are symmetric. Full article
(This article belongs to the Special Issue Advances in Graph Theory)
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15 pages, 675 KiB  
Article
Classification of Genus Three Zero-Divisor Graphs
by Thangaraj Asir, Karuppiah Mano and Turki Alsuraiheed
Symmetry 2023, 15(12), 2167; https://doi.org/10.3390/sym15122167 - 6 Dec 2023
Viewed by 1323
Abstract
In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by Γ(R), [...] Read more.
In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by Γ(R), is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if xy=0. Here, we classify the local rings with genus three zero-divisor graphs. Full article
(This article belongs to the Special Issue Advances in Graph Theory)
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11 pages, 4977 KiB  
Article
The Chromatic Entropy of Linear Supertrees and Its Application
by Feng Fu, Bo Deng and Liming Dai
Symmetry 2023, 15(11), 2061; https://doi.org/10.3390/sym15112061 - 14 Nov 2023
Viewed by 951
Abstract
Shannon entropy plays an important role in the field of information theory, and various graph entropies, including the chromatic entropy, have been proposed by researchers based on Shannon entropy with different graph variables. The applications of the graph entropies are found in numerous [...] Read more.
Shannon entropy plays an important role in the field of information theory, and various graph entropies, including the chromatic entropy, have been proposed by researchers based on Shannon entropy with different graph variables. The applications of the graph entropies are found in numerous areas such as physical chemistry, medicine, and biology. The present research aims to study the chromatic entropy based on the vertex strong coloring of a linear p-uniform supertree. The maximal and minimal values of the p-uniform supertree are determined. Moreover, in order to investigate the generalization of dendrimers, a new class of p-uniform supertrees called hyper-dendrimers is proposed. In particular, the extremal values of chromatic entropy found in the research for supertrees are applied to explore the behavior of the hyper-dendrimers. Full article
(This article belongs to the Special Issue Advances in Graph Theory)
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