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Graph Theory and Combinatorics: Theory and Applications

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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: 30 October 2025 | Viewed by 3572

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Special Issue Editors

Dr. Murong Xu

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Guest Editor

Department of Mathematics, University of Scranton, Scranton, PA 18510, USA
Interests: graph theory; combinatorics

Dr. Jian-Bing Liu

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Guest Editor

Department of Mathematics, University of Hartford, West Hartford, CT 06117, USA
Interests: Graph Theory; Topological Graph Theory

Prof. Dr. Xueliang Li

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Guest Editor

Center for Combinatorics, Nankai University, Tianjin 300071, China
Interests: graph theory and its applications; combinatorial optimizations; algorithms and complexity analysis; NP-hard problems; discrete mathematics and its applications in computer science, chemistry, biology, etc.
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Special Issue Information

Dear Colleagues,

Graph theory and combinatorics study discrete structures and their properties. Graph theory focuses on graphs, which are composed of vertices and edges, analyzing connectivity and structural patterns. Combinatorics deals with the counting, arrangements and combinations of discrete objects. Together, these fields provide essential tools for solving problems across diverse disciplines such as computer science, biology and optimization by analyzing relationships and developing efficient algorithms. Recently, there has been significant research activity in both graph theory and combinatorics, spanning theoretical advancements and practical applications.

We are pleased to announce a Special Issue dedicated to graph theory and combinatorics. We invite the submissions of original research articles, reviews and innovative applications within these fields. Manuscripts offering novel insights, methodologies or applications that contribute to advancing the understanding and application of graph theory and combinatorics are welcome.

Dr. Murong Xu
Dr. Jian-Bing Liu
Prof. Dr. Xueliang Li
Guest Editors

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. Axioms 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.

Keywords

graph theory
combinatorial optimization
digraphs
hypergraphs
connectivity
group connectivity
strongly group connectivity
strong arc connectivity
extremal problems
extremal digraphs
graph coloring
list coloring
planar graphs
nowhere-zero flows
edge coloring
strong edge coloring
chromatic number
decomposition
eigenvalues
supereulerian graphs
domination number

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Published Papers (5 papers)

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Research

19 pages, 643 KB

Open AccessArticle

MaxSum Spanning Tree Interdiction and Improvement Problems Under Weighted l_∞ Norm⁺

by Qiao Zhang, Junhua Jia and Xiao Li

Axioms 2025, 14(9), 691; https://doi.org/10.3390/axioms14090691 - 11 Sep 2025

Abstract

The Max+Sum Spanning Tree (MSST) problem, with applications in secure communication systems, seeks a spanning tree T minimizing

{max}_{e \in T} w (e) + \sum_{e \in T} c (e)

on a given edge-weighted undirected network [...] Read more.

The Max+Sum Spanning Tree (MSST) problem, with applications in secure communication systems, seeks a spanning tree T minimizing

{max}_{e \in T} w (e) + \sum_{e \in T} c (e)

on a given edge-weighted undirected network

G (V, E, c, w)

, where the sets V and E are the sets of vertices and edges, respectively. The functions c and w are defined on the edge set, representing transmission cost and verification delay in secure communication systems, respectively. This problem can be solved within

O (| E | log | V |)

time. We investigate its interdiction (MSSTID) and improvement (MSSTIP) problems under the weighted

l_{\infty}

norm. MSSTID seeks minimal edge weight adjustments (to either c or w) to degrade network performance by ensuring the optimal MSST’s weight is at least K, while MSSTIP similarly aims to enhance performance by making the optimal MSST’s weight at most K through minimal weight modifications. These problems naturally arise in adversarial and proactive performance enhancement scenarios, respectively, where network robustness or efficiency must be guaranteed through constrained resource allocation. We first establish their mathematical models. Subsequently, we analyze the properties of the optimal value to determine the relationship between the magnitude of a given number and the optimal value. Then, utilizing binary search methods and greedy techniques, we design four algorithms with time complexity

O (| E |^{2} log | V |)

to solve the above problems by modifying w or c. Finally, numerical experiments are conducted to demonstrate the effectiveness of the algorithms. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

37 pages, 5079 KB

Open AccessArticle

The Complexity of Classes of Pyramid Graphs Based on the Fritsch Graph and Its Related Graphs

by Ahmad Asiri and Salama Nagy Daoud

Axioms 2025, 14(8), 622; https://doi.org/10.3390/axioms14080622 - 8 Aug 2025

Viewed by 281

Abstract

A quantitative study of the complicated three-dimensional structures of artificial atoms in the field of intense matter physics requires a collaborative method that combines a statistical analysis of unusual graph features related to atom topology. Simplified circuits can also be produced by using similar transformations to streamline complex circuits that need laborious mathematical calculations during analysis. These modifications can also be used to determine the number of spanning trees required for specific graph families. The explicit derivation of formulas to determine the number of spanning trees for novel pyramid graph types based on the Fritsch graph, which is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle, is the focus of our study. We conduct this by utilizing our understanding of difference equations, weighted generating function rules, and the strength of analogous transformations found in electrical circuits. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

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14 pages, 320 KB

Open AccessArticle

Odd Cycles in Conditionally Faulty Enhanced Hypercube Networks

by Min Liu

Axioms 2025, 14(3), 201; https://doi.org/10.3390/axioms14030201 - 10 Mar 2025

Viewed by 547

Abstract

The n-dimensional enhanced hypercube

Q_{n, k} (1 \leq k \leq n - 1)

is a well-known variation of hypercube networks. Its structure can be obtained from the hypercube by adding

2^{n - 1}

complementary edges. We denote [...] Read more.

The n-dimensional enhanced hypercube

Q_{n, k} (1 \leq k \leq n - 1)

is a well-known variation of hypercube networks. Its structure can be obtained from the hypercube by adding

2^{n - 1}

complementary edges. We denote a network G to be a conditionally faulty model if every fault-free vertex of G connects at least two fault-free edges. Let

F_{v}

and

F_{e}

be the set of faulty vertices and faulty edges in

Q_{n, k} (1 \leq k \leq n - 1)

, respectively. In this paper, for the conditionally faulty

Q_{n, k}

with

| F_{v} | + | F_{e} | \leq 2 n - 5

, where

n (\geq 3)

and k have different parity, I prove that

Q_{n, k} - F_{v} - F_{e}

contains a fault-free cycle with every odd length l, where

n - k + 4 \leq l \leq 2^{n} - 2 | F_{v} | - 1

. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

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16 pages, 372 KB

Open AccessArticle

An O(kn)-Time Algorithm to Solve Steiner (k, k′)-Eccentricity on Trees

by Xingfu Li

Axioms 2025, 14(3), 166; https://doi.org/10.3390/axioms14030166 - 24 Feb 2025

Viewed by 378

Abstract

Steiner

(k, k^{'})

-eccentricity on a given fixed

k^{'}

-subset in a graph G is the maximum Steiner distance over all k-subsets of

V (G)

which contain the fixed

k^{'}

-set, where the Steiner [...] Read more.

Steiner

(k, k^{'})

-eccentricity on a given fixed

k^{'}

-subset in a graph G is the maximum Steiner distance over all k-subsets of

V (G)

which contain the fixed

k^{'}

-set, where the Steiner distance of a set is the size of a minimum Steiner tree on this set in a graph. Let

R \subseteq V (T)

be the given fixed

k^{'}

-subset in a tree T. Let

k_{1}

and

k_{2}

be two integers such that

k_{1} \geq k_{2} \geq k^{'}

. We prove that, in a tree, every optimal solution of Steiner

(k_{1}, k^{'})

-eccentricity on R takes some optimal solution of Steiner

(k_{2}, k^{'})

-eccentricity on R as a partial solution. On the other hand, every optimal solution of Steiner

(k_{2}, k^{'})

-eccentricity on R is part of some optimal solution of Steiner

(k_{1}, k^{'})

-eccentricity on the set R in a tree. Finally, we present an

O (k n)

-time algorithm to solve Steiner

(k, k^{'})

-eccentricity on a given fixed

k^{'}

-set in trees. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

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33 pages, 3247 KB

Open AccessArticle

Optimization of General Power-Sum Connectivity Index in Uni-Cyclic Graphs, Bi-Cyclic Graphs and Trees by Means of Operations

by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Axioms 2024, 13(12), 840; https://doi.org/10.3390/axioms13120840 - 28 Nov 2024

Cited by 1 | Viewed by 903

Abstract

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

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