Degree-Eccentricity Matrix of Graphs and Some Properties

This paper presents a new matrix for a given graph called the Degree-Eccentricity (DE) matrix, which consists of the degree and eccentricity of a vertex. Properties such as irreducibility and primitivity of this matrix are discussed. Further we obtain the spectrum and energy of DE matrices associated with various classes of graphs and some graphs obtained through graph operations. Also, we try to develop an algorithm to construct a new class of graph with DE energy equal to one. Further, we made an attempt to discover few graphs with DE energy equal to one. Also, an upper bound for the eigenvalues of DE matrix is obtained.