arXiv subjects

Kalpesh M. Popat

Publications and source records attributed to Kalpesh M. Popat.

6 recordsLinked to original sources

arXiv2026

Some New Results on Seidel Equienergetic Graphs

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance matrix, Common neighbourhood matrix and Seidel matrix. The Seidel matrix of the graph $G$ is the square matrix in which $ij^{th}$ entry is $-1$ or $1$, if the vertices $v_i$ and $v_j$ are adjacent or non-adjacent respectively, and is $0$ , if $v_i=v_j.$ The Seidel energy of $G$ is the sum of the absolute values of the eigenvalues of its Seidel matrix. We present here some graph families which are Seidel equienergetic.

math.CO↗

arXiv2026

Some New Results on Energy of Graphs with Self Loops

The graph $G_σ$ is obtained from graph $G$ by attaching self loops on $σ$ vertices. The energy $ E(G_σ)$ of the graph $G_σ$ with order $n$ and eigenvalues $λ_1,λ_2,\dots,λ_n$ is defined as $ E(G_σ)= \displaystyle \sum_{i=1}^n\left|λ_i-\dfracσ{n}\right| $. It has been proved that if $σ=0\; or\; n$ then $ E(G)=E(G_σ) $. The obvious question arise: Are there any graph such that $E(G)=E(G_σ)$ and 0$<σ<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.

math.CO↗

arXiv2026

Locally Equienergetic Graphs

For a given graph $ G $, let $ G^{(j)} $ denote the graph obtained by the deletion of vertex $ v_j $ from $ G $. The difference $ \mathscr{E}(G) - \mathscr{E}(G^{(j)}) $ quantifies the change in the energy of $ G $ upon the removal of $ v_j $, termed as the local energy of $ G $ at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by $\mathscr{E}_G(v)$. The local energy of the graph $ G $, therefore, is the summation of these vertex-specific local energies across all vertices in $ V(G) $, expressed by $ e(G) = \sum \mathscr{E}_G(v) $. Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs.

math.CO↗

arXiv2026

Graph Energies of Generalized and Shadow-Splitting Graphs

We extend the notions of the m-splitting graph Sm(G) and the m-shadow graph Dm(G) to introduce two new graph operations: the (p, q)-generalized splitting graph Sp,q(G) and the (c, k)-shadow-splitting graph Hc,k(G). We derive the adjacency energy of these constructions and as an application, identify several new infinite families of equienergetic and borderenergetic graphs.

math.CO↗

arXiv2026

On the Vertex Seidel Energy of Graphs

We introduce the vertex Seidel energy via the diagonal entries of the absolute Seidel matrix. We establish a spectral formula, compute exact values for several graph families, derive bounds, and present a Coulson-type integral representation for analytical study of this invariant. We also show that vertex Seidel energy is invariant under Seidel switching and complementation.

math.SP↗

arXiv2026

Some new results on the Seidel energy of graphs with self-loops

Harshitha et al. recently introduced Seidel energy of graphs with self loops. In this paper, we extend some of their results by giving a necessary and sufficient condition for the Seidel energy of a looped graph to be equal to the Seidel energy of its underlying graph. We also consider Seidel energy of the union of certain graphs, and show that graph operations complement and Seidel switching preserve Seidel energy in the looped setting.

math.GM↗