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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 31 August 2026 | Viewed by 3915

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Special Issue Editor

Dr. Randy R. Davila

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Guest Editor

Computational and Applied Mathematics Department, Rice University, Houston, TX 77005, USA
Interests: discrete mathematics; combinatorics; graph theory; applied mathematics

Special Issue Information

Dear Colleagues,

Graph theory and combinatorics remain central to modern mathematics, offering a deep well of unsolved problems, rich structural theory, and surprising connections to fields as varied as computer science, statistical physics, and optimization. This Special Issue is dedicated to exploring new perspectives in these areas, particularly topics that have seen recent growth in theoretical depth and computational insight.

We especially encourage submissions in the following areas: domination theory, including total, connected, Roman, and semi-total variants, as well as zero forcing, positive semidefinite zero forcing, power domination, and their connections to linear algebra, propagation processes, and control theory; independence number and related extremal and probabilistic problems; degree sequence problems, including characterizations, realizability questions, and invariants such as the residue of a graph; automated approaches to conjecture generation, including algorithmic heuristics that mimic mathematical discovery; new bounds, characterizations, and structural insights in classical and modern graph parameters; and interdisciplinary work linking combinatorics with algebraic, geometric, or logical methods.

While this Special Issue focuses firmly on the mathematical foundations of graph theory and combinatorics, we also welcome contributions that incorporate computational tools or explore how algorithmic processes can guide intuition and discovery in pure mathematics.

We invite original research articles and well-crafted surveys offering insight into ongoing developments, open problems, or novel frameworks.

Dr. Randy R. Davila
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 250 words) can be sent to the Editorial Office for assessment.

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

extremal graph theory
discrete mathematics
combinatorics
graph theory
applied mathematics
linear algebra
matrices
data science
degree sequence problems
zero forcing sets
zero forcing number
cardinalities

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Published Papers (5 papers)

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Research

12 pages, 268 KB

Open AccessArticle

Optimal Range of k-Consecutive Sums on a Circle for n = 2k + 1 and n = k² + 1

by Yaoran Yang and Yutong Zhang

Mathematics 2026, 14(8), 1252; https://doi.org/10.3390/math14081252 - 9 Apr 2026

Viewed by 128

Abstract

Arrange the integers

1, 2, \dots, n

on a circle and, for a fixed

k \geq 1

, let

s_{i}

be the sum of the k consecutive entries starting at position i (indices taken modulo n). For a [...] Read more.

Arrange the integers

1, 2, \dots, n

on a circle and, for a fixed

k \geq 1

, let

s_{i}

be the sum of the k consecutive entries starting at position i (indices taken modulo n). For a circular permutation

π

, define the range

R (π) = {max}_{i} s_{i} - {min}_{i} s_{i},

and let

w (n, k)

be the minimum value of

R (π)

over all circular permutations of

{1, \dots, n}

. We obtain three structural results. First, we prove the complement symmetry

w (n, k) = w (n, n - k)

. Second, we determine the first nontrivial arithmetic progression case

n = 2 k + 1

exactly:

w (2 k + 1, k) = 2 ⌈\frac{k}{2}⌉ .

Third, we determine the structured regime

n = k^{2} + 1

exactly:

w (k^{2} + 1, k) = k .

The proofs combine averaging lower bounds on the progression

n \equiv 1 (\mod k)

with explicit constructions: a parity-sensitive two-block arrangement for

n = 2 k + 1

and a

k \times k

array construction for

n = k^{2} + 1

. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

52 pages, 661 KB

Open AccessArticle

Graph-Theoretic Idealization of Semigroups via Bruck-Reilly Extensions

by Suha Wazzan and David A. Oluyori

Mathematics 2026, 14(5), 891; https://doi.org/10.3390/math14050891 - 5 Mar 2026

Viewed by 341

Abstract

This paper establishes a graph-theoretic framework for idealization semigroups arising from Bruck–Reilly extensions. Building on a recent study by Wazzan and Ozalan, we introduce five graph families—

Γ_{E}

Γ_{0}

Γ_{Cay}

Γ_{K}

, and [...] Read more.

Γ_{E}

Γ_{0}

Γ_{Cay}

Γ_{K}

, and

Γ (G_{k})

—each encoding a distinct algebraic facet of

S B_{i} (⋈) B

. We prove explicit correspondences linking combinatorial invariants to algebraic structure: diameter captures generating efficiency and semilattice height; girth signals short relations; chromatic number bounds idempotent cardinalities and

D

-class counts; clique number measures maximal commuting subsets; and Laplacian spectra encode ideal size and Schützenberger groups. Our central result demonstrates that Green’s relations are combinatorially recoverable from graph pairs. For commutative

S B_{i} (⋈) B

(Γ_{E}, Γ_{K})

uniquely determines

J

-order,

D

-classes, and

H

-classes via neighborhood inclusions, bipartite components, and automorphism orbits, yielding the first algorithmic reconstruction of ideal-theoretic structure from graph data. The framework is implemented in SageMath as a reproducible open-source toolkit validated on concrete examples. This work synthesizes algebraic graph theory, semigroup theory, and computational mathematics into a unified algebraic-combinatorial dictionary, providing both new analytical tools and a methodological template for studying algebraic constructions via graph invariants. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

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15 pages, 296 KB

Open AccessFeature PaperArticle

Local Energy of Digraphs

by Carlos Espinal and Juan Rada

Mathematics 2026, 14(4), 609; https://doi.org/10.3390/math14040609 - 10 Feb 2026

Cited by 1 | Viewed by 262

Abstract

Theenergy of a graph is a classical spectral invariant defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Recently, the notion of local energy was introduced to measure the contribution of vertices to the total energy via vertex deletion. In this paper, we first study the variation of graph energy under the deletion of a set of vertices and obtain general upper bounds in terms of vertex degrees, together with a characterization of the equality cases under natural structural conditions. These results provide the foundation for extending the concept of local energy to digraphs. Using the singular values of the adjacency matrix, we define the local energy of a digraph and derive sharp upper bounds in terms of the in-degree and out-degree of a vertex. The equality cases are characterized by introducing a special class of vertices, called star-vertices. Finally, we obtain sharp bounds for the total local energy of a digraph in terms of its energy and of the Randić index. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

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Figure 1

8 pages, 240 KB

Open AccessFeature PaperArticle

Some Problems and Conjectures About Voloshin Triple Systems

by Mario Gionfriddo

Mathematics 2026, 14(1), 42; https://doi.org/10.3390/math14010042 - 22 Dec 2025

Cited by 1 | Viewed by 455

Abstract

In this paper we give a short survey of Voloshin Triple Systems, which are Steiner Triple Systems with a vertex colouring which colour the blocks using exactly two colours. We also provide two conjectures about

V T S s

obtained by the [...] Read more.

V T S s

obtained by the construction

v ⟶ 2 v + 1

, starting from

v = 3

. Finally, we point out some open problems about Steiner Systems

S (2, 4, v)

with a similar vertex colouring, which could provide similar conjectures. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

17 pages, 340 KB

Open AccessFeature PaperArticle

Efficient Direct Reconstruction of Bipartite (Multi)Graphs from Their Line Graphs Through a Characterization of Their Edges

by Drago Bokal and Janja Jerebic

Mathematics 2025, 13(17), 2876; https://doi.org/10.3390/math13172876 - 5 Sep 2025

Viewed by 1093

Abstract

We study the line graphs of bipartite multigraphs, which naturally arise in combinatorics, game theory, and applications such as scheduling and motion planning. We introduce a new characterization of these graphs via valid partial assignments of the edges of the underlying bipartite multigraph to the vertices of its line graph. We show that an empty assignment extends to a complete one precisely when the graph is a line graph of a bipartite multigraph. Based on this, we design an

O (Δ (G) | E (G) |)

algorithm that incrementally constructs such assignments. The algorithm also provides a data structure supporting efficient solutions to problems of maximum clique, maximum weighted clique, minimum clique cover, chromatic number, and independence number. For line graphs of bipartite simple graphs these problems become solvable in linear time, improving on previously known polynomial-time results. For general bipartite multigraphs, our method enhances the

O (| V (G) |^{3})

recognition algorithm of Peterson and builds on the results of Demaine et al., Hedetniemi, Cook et al., and Gurvich and Temkin. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

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