Mathematics

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11 pages, 643 KB

Open AccessFeature PaperArticle

Interpolating Missing Spatial Data Using Graph Laplacian Eigenbasis

by Zihan Jin and Hiroshi Yamada

Mathematics 2026, 14(9), 1435; https://doi.org/10.3390/math14091435 - 24 Apr 2026

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This paper addresses the problem of interpolating missing spatial data at the vertices of a connected undirected simple graph. We show that, by exploiting the eigenbasis of the graph Laplacian, all missing values can be reconstructed even from a single observation. This work [...] Read more.

This paper addresses the problem of interpolating missing spatial data at the vertices of a connected undirected simple graph. We show that, by exploiting the eigenbasis of the graph Laplacian, all missing values can be reconstructed even from a single observation. This work establishes a novel connection between spatial statistics and spectral graph theory. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

16 pages, 1557 KB

Open AccessArticle

A Graph-Theoretical and Machine Learning Approach for Predicting Physicochemical Properties of Anti-Cancer Drugs

by Haseeb Ahmad and Alaa Altassan

Mathematics 2026, 14(6), 1003; https://doi.org/10.3390/math14061003 - 16 Mar 2026

Viewed by 373

Abstract

Topological graph theory provides a quantitative approach to understanding the structural complexities of sulfonamide compounds, which are prominent for their therapeutic importance in cancer treatment. A new computational scheme to predict the physicochemical and biological functions of sulfonamide derivatives, based on connection numbers [...] Read more.

Topological graph theory provides a quantitative approach to understanding the structural complexities of sulfonamide compounds, which are prominent for their therapeutic importance in cancer treatment. A new computational scheme to predict the physicochemical and biological functions of sulfonamide derivatives, based on connection numbers and connection-based topological indices as alternatives to the theoretically overt degree-based index, is proposed. A set of structurally diverse sulfonamide compounds as chemical graphs is considered, and the relevant graph descriptors are computed using different connection numbers. Due to the complexity of the calculations involved in connectivity and other such indices, algorithms were developed in Python 3.12.12 to automate the extraction and calculation of these indices. QSPR analysis, with the help of supervised machine learning models like linear regression, among others, and various statistical techniques, was employed to obtain insight into the relationships existing between the structural properties and the molecular properties measured, such as melting point, molecular weight, etc. These results demonstrate the great predictive capability of connection-based indices in assessing pharmacologic efficacy or molecular behavior. The holistic setting thus links topological modeling to data-driven prediction and provides a window into the rational design and optimization of sulfonamide-based cancer therapeutics. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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5 pages, 184 KB

Open AccessArticle

The Structure of Graphs Whose Vertices of Degree at Least Three Are at a Distance of at Least Three

by Salman Ghazal and Steve Karam

Mathematics 2026, 14(4), 637; https://doi.org/10.3390/math14040637 - 11 Feb 2026

Viewed by 588

Abstract

Let G be a connected graph in which the distance between any two distinct vertices of degree of at least three is at least three. We find the structure of G. It turns out that G decomposes into a tree and a [...] Read more.

Let G be a connected graph in which the distance between any two distinct vertices of degree of at least three is at least three. We find the structure of G. It turns out that G decomposes into a tree and a matching. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

13 pages, 764 KB

Open AccessArticle

On Even Vertex Magic Total Labelings of Plus Wheels and Some Wheel-Related Graphs

by Supaporn Saduakdee and Varanoot Khemmani

Mathematics 2026, 14(4), 583; https://doi.org/10.3390/math14040583 - 7 Feb 2026

Viewed by 451

Abstract

Let G be a graph with n vertices and m edges. A vertex magic total labeling of G is a bijection [...] Read more.

Let G be a graph with n vertices and m edges. A vertex magic total labeling of G is a bijection

f : V (G) \cup E (G) \to {1, 2, \dots, n + m}

such that, for each vertex

u \in V (G)

, the sum of the label of u and the labels of all edges incident to u is equal to a fixed constant, referred to as the magic constant. A vertex magic total labeling is said to be even if the labels assigned to the vertices are exactly even numbers

{2, 4, 6, \dots, 2 n}

. These labelings, along with related variations, have theoretical significance and practical applications, such as resource allocation, fault tolerance, and network design. Structured labelings aid channel assignment, address computation, and reduce collisions in networks. In this paper, we investigate wheel-related graphs that either admit or do not admit an even vertex magic total labeling. Furthermore, we introduce a new class of wheel-related graph, referred to as the plus wheel

W_{n} + r

, that can have such labelings, and we also establish a necessary and sufficient condition for such graphs to possess this property. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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35 pages, 504 KB

Open AccessArticle

Introducing a Resolvable Network-Based SAT Solver Using Monotone CNF–DNF Dualization and Resolution

by Gábor Kusper and Benedek Nagy

Mathematics 2026, 14(2), 317; https://doi.org/10.3390/math14020317 - 16 Jan 2026

Viewed by 669

Abstract

This paper is a theoretical contribution that introduces a new reasoning framework for SAT solving based on resolvable networks (RNs). RNs provide a graph-based representation of propositional satisfiability in which clauses are interpreted as directed reaches between disjoint subsets of Boolean variables (nodes). [...] Read more.

This paper is a theoretical contribution that introduces a new reasoning framework for SAT solving based on resolvable networks (RNs). RNs provide a graph-based representation of propositional satisfiability in which clauses are interpreted as directed reaches between disjoint subsets of Boolean variables (nodes). Building on this framework, we introduce a novel RN-based SAT solver, called RN-Solver, which replaces local assignment-driven branching by global reasoning over token distributions. Token distributions, interpreted as truth assignments, are generated by monotone CNF–DNF dualization applied to white (all-positive) clauses. New white clauses are derived via resolution along private-pivot chains, and the solver’s progression is governed by a taxonomy of token distributions (black-blocked, terminal, active, resolved, and non-resolved). The main results establish the soundness and completeness of the RN-Solver. Experimentally, the solver performs very well on pigeonhole formulas, where the separation between white and black clauses enables effective global reasoning. In contrast, its current implementation performs poorly on random 3-SAT instances, highlighting both practical limitations and significant opportunities for optimization and theoretical refinement. The presented RN-Solver implementation is a proof-of-concept which validates the underlying theory rather than a state-of-the-art competitive solver. One promising direction is the generalization of strongly connected components from directed graphs to resolvable networks. Finally, the token-based perspective naturally suggests a connection to token-superposition Petri net models. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

12 pages, 2043 KB

Open AccessFeature PaperArticle

On Vertex Magic 3-Regular Graphs with a Perfect Matching

by Tao-Ming Wang

Mathematics 2025, 13(24), 3969; https://doi.org/10.3390/math13243969 - 12 Dec 2025

Viewed by 1040

Abstract

Let

G = (V, E)

be a finite simple graph with

p = | V |

vertices and

q = | E |

edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from [...] Read more.

Let

G = (V, E)

be a finite simple graph with

p = | V |

vertices and

q = | E |

edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from

V \cup E

to the consecutive integers

1, 2, \dots, p + q,

with the property that, for every vertex

u \in V

, one has

f (u) + \sum_{u v \in E} f (u v) = k

for some magic constant k. The vertex magic total labeling is called E-super if furthermore

f (E) = {1, 2, \dots, q}

. A graph is called (E-super) vertex magic if it admits an (E-super) vertex magic total labeling. In this paper, we verify the existence of E-super vertex magic total labeling for a class of 3-regular graphs with a perfect matching, and we confirm the existence of such a labeling for general regular graphs of odd degree containing particular classes of 3-factors, which provides us with known and new examples. Note that Harary graphs are among the popular models used in communication networks. In 2012, G. Marimuthu and M. Balakrishnan raised a conjecture that if

n > 4

,

n \equiv 0 (m o d 4)

and m is odd, then the Harary graph

H_{m, n}

admits an E-super vertex magic labeling. Among others, we are able to verify this conjecture except for one case while

m = 3

and

n \equiv 4 (m o d 8)

. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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24 pages, 313 KB

Open AccessArticle

Asymptotics of Closeness Centralities of Graphs

by Santiago Frias, Adriana Galindo Silva, Bryan Romero and Darren A. Narayan

Mathematics 2025, 13(23), 3812; https://doi.org/10.3390/math13233812 - 27 Nov 2025

Viewed by 473

Abstract

Given a connected graph G with n vertices, the distance between two vertices is the number of edges in a shortest path connecting them. The sum of the distances in a graph G from a vertex v to all other vertices is denoted [...] Read more.

Given a connected graph G with n vertices, the distance between two vertices is the number of edges in a shortest path connecting them. The sum of the distances in a graph G from a vertex v to all other vertices is denoted by

S D_{G} (v)

. The closeness centrality of a vertex in a graph was defined by Bavelas to be

C_{C} (v) = \frac{n - 1}{S D_{G} (v)}

and the closeness centrality of G is

C_{C} (G) = \sum_{v \in G} \frac{n - 1}{S D_{G} (v)}

. We consider the asymptotic limit of

C_{C} (G)

as the number of vertices tends to infinity and provide an elegant and insightful proof of a 2025 result by Britz, Hu, Islam, and Tang,

lim_{n \to \infty} C_{C} (P_{n}) = π

, using uniform convergence and Riemann sums. We applied the same technique for the union of a cycle

C_{m}

and path

P_{n}

and the union of a path and a complete graph. We prove that of all graphs, paths have the minimum closeness centrality. Next we show for any

c \in [π, \infty)

, there exists a sequence of graphs

{G_{n}}

such that

lim_{n \to \infty} C_{C} (G_{n}) = c

. In addition, we investigate the mean distance of a graph,

\bar{l} (G) = \frac{1}{n (n - 1)} \sum_{v \in V (G)} S D (v)

and the normalized closeness centrality,

{\bar{C}}_{C} (G) = \frac{1}{n} C_{C} (G)

. We verify a conjecture of Britz, Hu, Islam, and Tang that the set of products

{\bar{l} (G) {\bar{C}}_{C} (G) : G is finite and connected}

is dense in

[1, 2)

. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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24 pages, 1525 KB

Open AccessArticle

Counting Tree-like Multigraphs with a Given Number of Vertices and Multiple Edges

by Muhammad Ilyas, Seemab Hayat and Naveed Ahmed Azam

Mathematics 2025, 13(21), 3405; https://doi.org/10.3390/math13213405 - 26 Oct 2025

Viewed by 1092

Abstract

The enumeration of chemical graphs plays a crucial role in cheminformatics and bioinformatics, especially in the search for novel drug discovery. These graphs are usually tree-like multigraphs, or they consist of tree-like multigraphs attached to a central core. In both configurations, the tree-like [...] Read more.

The enumeration of chemical graphs plays a crucial role in cheminformatics and bioinformatics, especially in the search for novel drug discovery. These graphs are usually tree-like multigraphs, or they consist of tree-like multigraphs attached to a central core. In both configurations, the tree-like components play a key role in determining the properties and activities of chemical compounds. In this work, we propose a dynamic programming approach to precisely count the number of tree-like multigraphs with a given number of n vertices and Δ multiple edges. Our method transforms multigraphs into rooted forms by designating their unicentroid or bicentroid as the root and then defining a canonical representation based on the maximal subgraphs rooted at the root’s children. This canonical form ensures that each multigraph is counted only once. Recursive formulas are then established based on the number of vertices and multiple edges in the largest subgraphs rooted at the root’s children. The resulting algorithm achieves a time complexity of

O (n^{2} (n + Δ (n + Δ^{2} \cdot min {n, Δ})))

and space complexity of

O (n^{2} (Δ^{3} + 1))

. Extensive experiments demonstrate that the proposed method scales efficiently, being able to count multigraphs with up to 200 vertices (e.g., (200, 26)) and up to 50 multiple edges (e.g., (90, 50)) in under 15 min. In contrast, the available state-of-the-art tool Nauty runs out of memory beyond moderately sized instances, as it relies on explicit generation of all candidate multigraphs. These results highlight the practical advantage and strong potential of the proposed method as a scalable tool for chemical graph enumeration in drug discovery applications. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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19 pages, 7042 KB

Open AccessFeature PaperArticle

Graph Theoretic Analyses of Tessellations of Five Aperiodic Polykite Unitiles

by John R. Jungck and Purba Biswas

Mathematics 2025, 13(18), 2982; https://doi.org/10.3390/math13182982 - 15 Sep 2025

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Abstract

Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary objective here is to learn how we could build a [...] Read more.

Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary objective here is to learn how we could build a self-assembling polyhedron that would have an aperiodic tessellation of its surface using only a single type of polykite unitile. Such a structure would be analogous to some viral capsids that have been reported to have a quasicrystal configuration of capsomeres. We report on our use of a graph–theoretic approach to examine the adjacency and symmetry constraints of these unitiles in tessellations because by using graph theory rather than the usual geometric description of polykite unitiles, we are able (1) to identify which particular vertices and/or edges join one another in aperiodic tessellations; (2) to take advantage of being scale invariant; and (3) to use the deformability of shapes in moving from the plane to the sphere. We systematically classify their connectivity patterns and structural characteristics by utilizing Hamiltonian cycles of vertex degrees along the perimeters of the unitiles. In addition, we applied Blumeyer’s 2 × 2 classification framework to investigate the influence of chirality and periodicity, while Heesch numbers of corona structures provide further insights into tiling patterns. Furthermore, we analyzed the distribution of polykite unitiles with Voronoi tessellations and their Delaunay triangulations. The results of this study contribute to a better understanding of self-assembling structures with potential applications in biomimetic materials, nanotechnology, and synthetic biology. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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11 pages, 279 KB

Open AccessArticle

A Vertex Separator Problem for Power Graphs of Groups

by Haeder Younis Althoby, Mohammed A. Mutar and Daniele Ettore Otera

Mathematics 2025, 13(18), 2970; https://doi.org/10.3390/math13182970 - 14 Sep 2025

Cited by 1 | Viewed by 963

Abstract

Given a graph G, an st-connected vertex separator (CVS) problem refers to the search for a minimum connected component in G whose removal leaves the pair of nodes s and t in two disjoint components. We investigate this specific problem on certain [...] Read more.

Given a graph G, an st-connected vertex separator (CVS) problem refers to the search for a minimum connected component in G whose removal leaves the pair of nodes s and t in two disjoint components. We investigate this specific problem on certain types of graphs—so-called power graphs associated with groups. We present a mathematical model and an algorithm to solve this problem in a reasonable time. Finally, numerical tests show that the algorithm runs considerably fast even on graphs of high orders. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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27 pages, 2240 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Hybrid Entropy-Based Metrics for k-Hop Environment Analysis in Complex Networks

by Csaba Biró

Mathematics 2025, 13(17), 2902; https://doi.org/10.3390/math13172902 - 8 Sep 2025

Cited by 1 | Viewed by 1000

Abstract

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when [...] Read more.

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when classical indices (e.g., degree or clustering) are similar. The metrics are formally defined and shown to be bounded, isomorphism-invariant, and stable under small edge edits. Their behavior is assessed on representative topologies (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, random geometric graphs, and the Zephyr quantum architecture). Across these settings, EWR and NED display predominantly negative correlation with degree and provide information largely orthogonal to standard centralities; vertices with identical degree can differ by factors of two to three in the proposed scores, revealing bridges and heterogeneous regions. These properties indicate utility for vulnerability assessment, topology-aware optimization, and layout heuristics in engineered and quantum networks. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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15 pages, 240 KB

Open AccessFeature PaperArticle

The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs

by Rao Li

Mathematics 2025, 13(17), 2897; https://doi.org/10.3390/math13172897 - 8 Sep 2025

Viewed by 988

Abstract

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian [...] Read more.

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, we first establish inequalities on the first Zagreb index and the Laplacian spectral radius of a graph. Using the ideas of proving the inequalities, we present sufficient conditions involving the first Zagreb index and the Laplacian spectral radius for some Hamiltonian properties of graphs. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

13 pages, 271 KB

Open AccessFeature PaperArticle

Modular H-Irregularity Strength of Graphs

by Martin Bača, Marcela Lascsáková and Andrea Semaničová-Feňovčíková

Mathematics 2025, 13(16), 2599; https://doi.org/10.3390/math13162599 - 14 Aug 2025

Viewed by 792

Abstract

Two new graph characteristics, the modular edge H-irregularity strength and the modular vertex H-irregularity strength, are introduced. Lower bounds on these graph characteristics are estimated, and their exact values are determined for certain families of graphs. This demonstrates the sharpness of [...] Read more.

Two new graph characteristics, the modular edge H-irregularity strength and the modular vertex H-irregularity strength, are introduced. Lower bounds on these graph characteristics are estimated, and their exact values are determined for certain families of graphs. This demonstrates the sharpness of the presented lower bounds. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

9 pages, 1015 KB

Open AccessArticle

Extremal Values of Second Zagreb Index of Unicyclic Graphs Having Maximum Cycle Length: Two New Algorithms

by Hacer Ozden Ayna

Mathematics 2025, 13(15), 2475; https://doi.org/10.3390/math13152475 - 31 Jul 2025

Viewed by 905

Abstract

It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could be stated as the condition that [...] Read more.

It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could be stated as the condition that the order and the size of the graph are equal. Using a recent result saying that the length of the unique cycle could be any integer between 1 and n − a₁ where a₁ is the number of pendant vertices in the graph, two explicit labeling algorithms are provided that attain these extremal values of the first and second Zagreb indices by means of an application of the well-known rearrangement inequality. When the cycle has the maximum length, we obtain the situation where all the pendant vertices are adjacent to the support vertices, the neighbors of the pendant vertices, which are placed only on the unique cycle. This makes it easy to calculate the second Zagreb index, as the contribution of the pendant edges to such indices is fixed, implying that we can only calculate these indices for the edges on the cycle. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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