Relationship between adjacency and distance matrix of graph of diameter two

Abstract

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0.  In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is D=2(J-I)-A. From this relationship, we  also determine the value of the determinant matrix A+D and the upper bound of determinant of matrix D.