This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices to contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.
Abstract
The success of Spectral Graph Theory is due to its immense applications in several fields of science and technology. The parameter `graph energy' plays a significant role in the Hückel Molecular Orbital Theory, where it is used to approximate the total π -electron energy of conjugated hydrocarbons. Several modified versions of the energy parameter have been defined by many researchers depending on the context. This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices. When the adjacency matrix provides information only on the existence of a relation among vertex pairs, these refined matrices give more information and weightage to the relation. These matrix-based energies quantify the total spectral energy of a system and provide insight into how structural modifications affect global stability and information distribution. The characteristic polynomials and corresponding eigenvalue spectra of the central graph are obtained for each matrix type, revealing how the spectral measures evolve under structural transformation. The analysis is further extended to the Indu–Bala product of graphs, a non-commutative operation that intricately merges the structural components of two graphs. The explicit expressions for the characteristic polynomial and spectral energies are derived, illustrating the effect of graph composition on spectral distribution. Additionally, for the operations of double-duplication and duplication vertex-join, the closed-form relations for the Zagreb and sum-connectivity spectra and their corresponding energy values are presented. The obtained results contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.
Chemical graph theory is essential for deriving graph spectral radius, especially in quantum chemistry; a significant link exists between eigenvalues and this invariant of spectral graph theory. This invariant is derived from spectrum, and has numerous applications in various fields and is used to solve many real-life problems. Zagreb-type indices and their spectrum variants are crucial for QSAR/QSPR modeling, chemoinformatics, and computational drug design [16]. The literature contains numerous variants of graph spectral radius. In this article, we concentrate on a novel extension of the spectral invariant called Second Hyper Zagreb spectral radius, which is obtained by replacing the adjacency matrix with the Second Hyper Zagreb matrix. The two graph operations that are primarily covered in this article are splitting and shadow graphs. The connection between the Second Hyper Zagreb spectral radius of these graph operations and Second Hyper Zagreb spectral radius of base graph Gis of special relevance to us. By using these operations, we can address the challenge of figuring out the relationship between the spectral radius of the base graph Gand the spectral radius of the newly generated graph. The findings of this article improve our understanding and pave the way for future research in spectral graph theory, where graph operations are frequently employed and play an important role in solving real-world problems.
A. Bilal, Muhammad Mobeen Munir· Scientific Annals of Compute...· 0 citations
This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Tur\'{a}n graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called \emph{the graphs of pyramids}, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lov\'{a}sz $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.
The ‘divide and conquer’ paradigm proves to be one of the most frequently used techniques for dealing with the complexities of graph-related problems. Therefore, it is of great importance to measure the tendency of a vertex to be critical and its susceptibility in a graph. The criticality of a vertex is often analysed in terms of its strength. Removing a highly critical vertex from a graph modelling a network may introduce vulnerability into the system represented by the graph. Minimizing the vulnerability of such a network without affecting its fundamental structure, thereby improving the stability of the graph, is the primary objective of the article. To achieve this, certain properties of Euler graphs are analysed in terms of vertex strength, and a method is presented for determining all possible constructions of Euler graphs corresponding to different integer partitions. The parts of a partition represent the vertex strengths, and their sum corresponds to the total vertex strength of the graph. Various connectivity indices are employed to validate the proposed constructions. Furthermore, their interrelationships and potential real-life applications are also discussed. It is evident from the constructions that they may play a vital role in developing network deception technology to protect digital assets, as each partition of the network generates a distinct network.
Saifur Rahman, Raju Doley· Acta Universitatis Sapientia...· 0 citations
This paper presents a comparative analysis of two variants of a classical algorithm for finding the shortest path in a connected graph. The first variant uses an adjacency matrix (AM) to verify the existence of an edge (arc) between two vertices, while the second variant performs the same verification using an adjacency list (AL). The objective of this study is to examine how graph density affects the performance of the two algorithmic modifications depending on the data structure used. A total of 95 graphs were analyzed, grouped into five sets from 100 to 500 in increments of 100. For each group, 19 graphs were generated with densities ranging from 5% to 95% in increments of 5%. The methodology includes analyzing the number of iterations, assignments, and comparisons executed by the algorithms for all graphs. The initial hypothesis assumed that the total number of operations would always be lower when using an AL instead of an adjacency matrix, regardless of graph density. The results demonstrate that this assumption is incorrect: for densities above 82%, the total number of operations is lower when using an adjacency matrix, whereas the AL is more efficient for densities below 82%, with its efficiency increasing as density decreases. These findings are particularly important for mobile technologies, as they support the design of more efficient pathfinding solutions that optimize performance and energy consumption in mobile applications.
V. Kralev, Radoslava Kraleva, Aleksandra Popova· International Journal of Int...· 0 citations
This study investigates the path energy bounds of hub-centric graph families of order (2n+1), denoted by ℋ2n+1. The path energy is defined as the absolute sum of the eigenvalues of the path adjacency matrix Ap(ℋ2n+1), where each entry pij measures the maximum number of internally vertex- disjoint paths. Through an in-depth examination of the characteristic values of this matrix Ap(ℋ2n+1), we derive path energy bounds specific to these four (2n+1)-vertex graph, namely closed helm, double star, friendship, and double wheel graphs. We further developed an explicit spectral construction enabling the numerical validation algorithm to evaluate the time complexity in various structural configurations of (2n+1)-graphs. Also executed extensive trials to capture their average, maximum, and minimum computational performances. This analysis offers a detailed comparative study of the structural attributes that lead to computational complexity, highlighting the substantial algorithmic demands for several graphs.
R. Jerlinkasmir, J. Veninstine Vivik, I. N. Cangul et al.· Scientific Annals of Compute...· 0 citations
Graph energy is an important concept in spectral graph theory with applications in mathematics and chemistry. In this paper, we study the Laplacian minimum domination energy of derived graphs of some standard graphs. The main aim is to obtain formulas, properties, and bounds for this energy measure. The study considers derived graphs of star graphs, complete bipartite graphs, friendship graphs, and healthy spider graphs. Using minimum dominating sets, minimum domination adjacency matrices, and Laplacian minimum domination matrices, the eigenvalues of these derived graphs are determined. Based on these eigenvalues, explicit formulas for the Laplacian minimum domination energy are obtained. Further, some basic properties related to eigenvalues are established. Upper and lower bounds for the Laplacian minimum domination energy are also derived using matrix methods and classical inequalities such as the Cauchy-Schwarz inequality. The results extend existing work on graph energy by combining domination concepts, Laplacian matrices, and derived graphs. The formulas, properties, and bounds obtained in this paper provide a better understanding of the spectral behavior of derived graphs and may be useful for further research in graph theory and its applications.
Jagadeesh Rajanna, Ashwini Ankanahalli Shashidhara· American Journal of Applied...· 0 citations