Adjacency-diametrical matrix of a graph

TL;DR

This paper introduces the adjacency-diametrical matrix of a graph, analyzes its spectrum for various graph classes, and explores its relation to graph invariants and graph operations.

Contribution

It determines the spectrum of the AD matrix for specific graph families and characterizes bipartite graphs using AD matrix properties, providing new insights into graph spectra.

Findings

Spectrum of AD matrix for paths, cycles, double star graphs

Determinant of AD matrix for connected graphs

Characterization of bipartite graphs via AD matrix eigenvalues

Abstract

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if $d_G(v_i,v_j)=d$, and $0$ otherwise, where $d_G(v_i,v_j)$ denotes the distance between the vertices $v_i$ and $v_j$ in $G$. We determine the spectrum of the AD matrix for paths, cycles, and double star graphs and obtain its determinant for a connected graph. We characterize a class of bipartite graphs using the coefficients of the characteristic polynomial and the eigenvalues of the AD matrix. We establish bounds relating the eigenvalues of the AD matrix to various graph invariants, and we determine the spectrum of the AD matrix for graphs formed by the join, lexicographic product, and Cartesian product operations under certain conditions on the constituent graphs.