Jul 2026· American Journal of Applied Mathematics· Vol 14, pp. 210-219· 0 citations· 6 references
Abstract
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.
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.
Let 𝐺 be a 𝑘-regular graph on 𝑛 vertices. In this paper, we determine the maximum and minimum degree energies of two specific graph operations: the extended 𝑚-splitting graph. 𝑆𝑝𝑙𝑚∗ (𝐺) and the 𝑚-semi shadow graph 𝑆𝐷𝑚(𝐺). By applying block matrix decompositions and unitary similarity transformations, we express the maximum and minimum degree spectra of these constructs explicitly in terms of the ordinary adjacency spectrum of the base graph 𝐺. Furthermore, we derive exact, closed-form expressions for their respective maximum and minimum degree energies. As applications of our main theorems, the corresponding energies for well-known families of regular graphs—including cycle, complete, and complete bipartite graphs—are explicitly established.
Dipakkumar N. Barad, V. Kaneria, K. Popat· International Journal of Mat...· 0 citations
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.
S. Sripriya, A. Anuradha· Baghdad Science Journal· 0 citations
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Vishal Gupta, A. Iosevich, J. Iosevich et al.· 0 citations
Consider a group [Formula: see text] and construct its power graph, whose vertex set consists of the elements of [Formula: see text]. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group [Formula: see text], the dihedral group [Formula: see text], and the dicyclic group [Formula: see text]. For [Formula: see text] the power graph of the cyclic group [Formula: see text] is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups
Priti Prasanna Mondal, Basit A. Mir, Fouzul Atik· Journal of Algebra and its A...· 0 citations
This study establishes exact formulas for the energy, Laplacian energy, and signless Laplacian energy of the lintang graph Ln and its line graph L(Ln). An algebraic spectral framework is employed to construct the associated matrices and determine their eigenvalues explicitly. In contrast to general results on line graph energies introduced by Ivan Gutman and subsequent studies, this work focuses on a specific two-hub graph structure for which explicit spectral formulas have not been previously reported. The results show that E(Ln) = 2 sqrt(2n), LE(L1) = QE(L1) = 10 sqrt(3), LE(Ln) = QE(Ln) = 4(n^2 - n + 2)/(n + 2) for n >= 2, and E(L(Ln)) = LE(L(Ln)) = QE(L(Ln)) = 4n - 4. These findings reveal a transition from sublinear to linear growth under the line graph transformation. The results contribute to spectral graph theory and are relevant to chemical graph theory and network analysis.
F. Fran, Meliana Pasaribu, Helmi Helmi et al.· ZERO Jurnal Sains Matematika...· 0 citations
