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Special Issue "Spectral Graph Theory, Molecular Graph Theory and Their Applications"

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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 3228

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Dear Colleagues,

Graph Theory is the area of mathematics that studies networks or graphs. It arose from the need to analyze many diverse network-like structures like the internet, molecules, road networks, social networks, education networks, and electrical networks. Spectral graph theory (a branch of graph theory) deals with the connection between the eigenvalues of a matrix associated with a graph and the corresponding structure of the graph. The first practical need for studying graph eigenvalues was in quantum chemistry in the nineteen-thirties, forties, and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. Several special kinds of graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix, etc.) are very popular in spectral graph theory.

A graphical invariant is a function from the set of graphs to the set of reals which is invariant under graph automorphisms. In chemical graph theory, graphical invariants are most often referred to as topological indices. Among the oldest and most well-known topological indices, Wiener index, Randić index, and Zagreb indices, etc. were introduced in the literature. In the last twenty years, a large number of mathematical investigations were reported on graph invariants (topological indices) whose origin is in chemistry, and which are claimed to have chemical applications.

The aim of this Special Issue is to publish original research articles focusing on spectral graph theory and the topological indices of graphs, as well as applications in these areas. To find new connections between the eigenvalues and eigenvectors of graphs and graph theory, parameters such as average degree, maximum degree, cliques, chromatic number, and matching number are welcome. In addition, this Special Issue covers the characterization of extremal graphs with respect to the existing popular topological indices of graphs with fixed graph parameters.

Prof. Dr. Kinkar Chandra Das
Guest Editor

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Keywords

spectral graph theory
topological indices of graphs
energy of graphs
combinatorics
extremal graph theory
graph polynomials

Published Papers (3 papers)

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Open AccessArticle

A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

and

Axioms 2023, 12(7), 675; https://doi.org/10.3390/axioms12070675 - 09 Jul 2023

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Abstract

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for

α \in R^{+} \ {1} .

Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for

α \in R^{+} \ {1},

we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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Open AccessArticle

On Graphs with c₂-c₃ Successive Minimal Laplacian Coefficients

Yue Xu

and

Shi-Cai Gong

Axioms 2023, 12(5), 464; https://doi.org/10.3390/axioms12050464 - 11 May 2023

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Abstract

Let G be a graph of order n and

L (G)

be its Laplacian matrix. The Laplacian polynomial of G is defined as [...] Read more.

Let G be a graph of order n and

L (G)

be its Laplacian matrix. The Laplacian polynomial of G is defined as

P (G; λ) = det (λ I - L (G)) = \sum_{i = 0}^{n} {(- 1)}^{i} c_{i} (G) λ^{n - i}

, where

c_{i} (G)

is called the i-th Laplacian coefficient of G. Denoted by

G_{n, m}

the set of all

(n, m)

-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each

i (i = 0, 1, \dots, n)

, H is

c_{i} (G)

-minimal in

G_{n, m}

. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in

G_{n, m}

because such graphs have minimal Laplacian-like energy. We determine that the

c_{2} (G)

c_{3} (G)

successive minimal graph is exactly one of the four classes of threshold graphs. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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Open AccessArticle

Extremal Graphs to Vertex Degree Function Index for Convex Functions

and

Axioms 2023, 12(1), 31; https://doi.org/10.3390/axioms12010031 - 27 Dec 2022

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Abstract

The vertex-degree function index

H_{f} (Γ)

is defined as

H_{f} (Γ) = \sum_{v \in V (Γ)} f (d (v))

for a function

f (x)

defined on non-negative [...] Read more.

The vertex-degree function index

H_{f} (Γ)

is defined as

H_{f} (Γ) = \sum_{v \in V (Γ)} f (d (v))

for a function

f (x)

defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for

H_{f} (Γ) + H_{f} (\bar{Γ})

and

H_{f} (Γ) \cdot H_{f} (\bar{Γ})

. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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