Adjacency-diametrical matrix of a graph
S. P. Leka Amruthavarshini, R. Rajkumar
TL;DR
This work introduces the adjacency-diametrical matrix $AD(G)=A(G)+d\,A_d(G)$ for a connected graph $G$ with diameter $d$ and studies its spectral properties through a weighted-graph viewpoint via $\mathcal{G}_G$. It provides exact AD-spectra for fundamental graph families (paths, cycles, double stars) and a determinant formula expressed via spanning adjacency-diametrical partitions, plus parity-based results and a characterization of diametrical bipartite graphs. The paper further derives bounds linking AD-eigenvalues to elementary graph invariants and investigates how AD-spectra behave under graph products (join, lexicographic, Cartesian). These results connect diameter-driven structure to spectral data, offering tools for graph classification and product-based spectral analysis with potential applications in network analysis and chemistry.
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.