Distance Seidel matrix of a connected graph
Haritha T, Chithra A.
TL;DR
This work introduces the distance Seidel matrix $\mathcal{D}^S(G)= J - I - 2\mathcal{D}(G)$ for connected graphs, establishing ties between $\mathcal{D}^S$-eigenvalues and the classical distance and adjacency spectra. It derives structural and spectral properties, including a characterization of graphs with $\partial_1^S=3$, and provides tight bounds for the spectral radius and energy $E_{\mathcal{D}^S}(G)$, together with exact values for several standard graphs. The paper also analyzes how $\mathcal{D}^S$-spectrum behaves under graph operations (join, products, double and extended double cover), and presents families of distance Seidel cospectral and integral graphs, enriching the spectrum-structural toolkit for distance-based Seidel-type matrices. These results enhance understanding of how distance and Seidel-type constructions interact spectrally, with implications for graph characterization and graph energy theory.
Abstract
For a connected graph $G$, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix $\mathcal{D}^S(G)$. Suppose that the eigenvalues of $\mathcal{D}^S(G)$ be $\partial_{1}^{S}(G) \geq \cdots \geq \partial_{n}^{S}(G).$ In this article, we establish a relationship between distance Seidel eigenvalues of a graph with its distance and adjacency eigenvalues. We characterize all the connected graphs with $\partial_{1}^{S}(G)= 3.$ Also, we determine different bounds for the distance Seidel spectral radius and distance Seidel energy. We study the distance Seidel energy change of the complete bipartite graph due to the deletion of an edge. Moreover, we obtain the distance Seidel spectra of different graph operations such as join, cartesian product, lexicographic product, and unary operations like the double graph and extended double cover graph. We give various families of distance Seidel cospectral and distance Seidel integral graphs as an application.