The relation between the square of the adjacency matrix and spectra of the distance matrix of a graph with diameter two

A graph is a set of vertices and edges where each edge connects two vertices in the graph. There are several ways to represent a graph, e.g., it can be represented as a matrix: its adjacency matrix, anti-adjacency matrix, Laplacian matrix, or distance matrix. Two matrices which will be explored in this paper are the adjacency matrix and distance matrix. The adjacency matrix represents the presence or absence of an edge connecting two vertices, while the distance matrix represents the shortest path between two vertices. A graph with diameter two is a graph such that the longest distance between any two vertices is equal to two. Examples of two-diameter graphs include bipartite graphs, wheel graphs, and fan graphs. The relation between the distance and adjacency matrices is already known. The purpose of this paper is to find the relation between the square of the distance matrix and the square of the adjacency matrix of a two diameter graph and to find the characteristic polynomial of the distance matrix of a complete bipartite graph Kn,n (which is a special type of two-diameter graph). We prove that where D2=4(J−I)A−+A2 where D is the distance matrix, J is the matrix all of whose entries are 1, I is the identity matrix, A is the adjacency matrix and Ā is the complement of A. Moreover, we also find that the characteristic polynomial of the distance matrix of a complete bipartite graph is P(λ)=(λ+2)2n−2(λ−(n−2))(λ−(3n−2)).