Eccentricity matrix of corona of two graphs

Abstract

The eccentricity matrix, ɛ(G), of a graph G is derived from the distance matrix by letting the uv-th element to be equal to the distance between two vertices u and v, if the distance is the minimum of their eccentricities and zero otherwise. In this article, we study the spectrum of ɛ(G) and establish an upper bound for its ɛ-spectral radius when G is a self-centered graph. Further, we explore the structure of ɛ(G∘H), where G∘H is the corona product of a self-centered graph G and a graph H. We characterize the irreducibility of ɛ(G∘H) and, in this process, find that it is independent of ɛ(H), which allows us to construct infinitely many graphs with irreducible eccentricity matrix. Moreover, we compute the complete spectrum of ɛ(G∘H) including its ɛ-eigenvectors, ɛ-energy, and ɛ-inertia. Finally, we conclude that several non-isomorphic ɛ-co-spectral graphs can be generated using the corona product of two graphs. © 2024 Elsevier B.V.