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Advances in Graph Theory Ⅱ

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Dr. Yuefang Sun

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School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
Interests: graph theory; extremal combinatorics; combinatorial optimization; algorithms and complexity analysis
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Special Issue Information

Dear Colleagues,

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

The aim of this Special Issue is to collect original research articles on this subject. We welcome the submission of papers that present new results related to all aspects of graph theory, particularly symmetric phenomena. This Special Issue is a continuation of the following Special Issue: mdpi.com/si/182093

The scope of this Special Issue includes, but is not limited to, the following topics:

Structural graph theory;
Extremal graph theory;
Random graph theory;
Spectral and algebraic graph theory;
Chemical graph theory;
Topological graph 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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Published Papers (3 papers)

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Research

10 pages, 222 KiB

Open AccessArticle

Maximum Colored Cuts in Edge-Colored Complete k-Partite Graphs and Complete Graphs

by Huawen Ma

Symmetry 2025, 17(5), 790; https://doi.org/10.3390/sym17050790 - 20 May 2025

Abstract

The Maximum Colored Cut problem aims to seek a bipartition of the vertex set of a graph, maximizing the number of colors in the crossing edges. It is a classical Max-Cut problem if the host graph is rainbow. Let [...] Read more.

m c c (G)

denote the maximum number of colors in a cut of an edge-colored graph G. Let

C_{k}

be a cycle of length k; we say G is PC-

C_{k}

-free if G contains no properly colored

C_{k}

. We say G is a p-edge-colored graph if there exist p colors in G. In this paper, we first show that if G is a PC-

C_{3}

-free p-edge-colored complete 4-partite graph, then

m c c (G) = p

. Let

k \geq 3

be an integer. Then, we show that if G is a PC-

C_{4}

-free p-edge-colored complete k-partite graph, then

m c c (G) \geq min {p - 1, 15 p / 16}

. Finally, for a p-edge-colored complete graph G, we prove that

m c c (G) \geq p - 1

if G is PC-

C_{4}

-free, and

m c c (G) \geq min {p - 6, 7 p / 8}

if G is PC-

C_{5}

-free and

p \geq 7

. Full article

(This article belongs to the Special Issue Advances in Graph Theory Ⅱ)

28 pages, 792 KiB

Open AccessArticle

Optimizing Decision-Making Using Domination Theory in Product Bipolar Fuzzy Graphs

by Wei Ming, Areen Rasool, Umar Ishtiaq, Sundas Shahzadi, Mubariz Garayev and Ioan-Lucian Popa

Symmetry 2025, 17(4), 479; https://doi.org/10.3390/sym17040479 - 22 Mar 2025

Viewed by 212

Abstract

The bipolar fuzzy model is a rapidly evolving research area that provides a robust framework for addressing real-world problems, with wide-ranging applications in scientific and technical domains. Within this framework, bipolar fuzzy graphs play a significant role in decision-making and problem-solving, particularly through domination theory, which helps tackle practical challenges. This study explores various operations on product bipolar fuzzy graphs, including union (∪), join (+), intersection (∩), Cartesian product (×), composition (∘), and complement, leading to the generation of new graph structures. Several important results related to complete product bipolar fuzzy graphs under these operations are established. Additionally, we introduce key concepts such as dominating sets, minimal dominating sets, and the domination number

⋎ (H)

, supported by illustrative examples. This study further investigates the properties of domination in the context of these operations. To demonstrate practical applicability, we present a decision-making problem involving the optimization of bus routes and the strategic placement of bus stations using domination principles. This research contributes to the advancement of bipolar fuzzy graph theory and its practical applications in real-world scenarios. Full article

(This article belongs to the Special Issue Advances in Graph Theory Ⅱ)

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11 pages, 288 KiB

Open AccessArticle

The Generalized Characteristic Polynomial of the K_m,n-Complement of a Bipartite Graph

by Weiliang Zhao and Helin Gong

Symmetry 2025, 17(3), 328; https://doi.org/10.3390/sym17030328 - 21 Feb 2025

Viewed by 429

Abstract

The generalized matrix of a graph G is defined as

M (G) = A (G) - t D (G)

(

t \in R

, and

A (G)

and

D (G)

, respectively, denote [...] Read more.

The generalized matrix of a graph G is defined as

M (G) = A (G) - t D (G)

(

t \in R

, and

A (G)

and

D (G)

, respectively, denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of

M (G)

. Let

K_{m, n}

be the complete bipartite graph. Then, the

K_{m, n}

-complement of a subgraph G in

K_{m, n}

is defined as the graph obtained by removing all edges of an isomorphic copy of G from

K_{m, n}

. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the

K_{m, n}

-complement of a bipartite subgraph G is provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the

K_{m, n}

-complements of G is given. Full article

(This article belongs to the Special Issue Advances in Graph Theory Ⅱ)

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