Distance Seidel matrix of a connected graph

TL;DR

This paper introduces the distance Seidel matrix for connected graphs, explores its eigenvalues, and investigates spectral properties, bounds, and effects of graph operations, providing new insights into graph spectra related to distance and Seidel matrices.

Contribution

It defines the distance Seidel matrix, relates its eigenvalues to known spectra, characterizes graphs with specific eigenvalues, and analyzes spectral properties under various graph operations.

Findings

Characterized graphs with maximum eigenvalue 3.

Derived bounds for spectral radius and energy.

Determined spectra for graph operations and identified cospectral and integral graphs.

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.