Adjacency-diametrical matrix of a graph

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.