Eccentricity matrix of corona of two graphs

Abstract

The eccentricity matrix, $\varepsilon \left(G\right)$, 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 $\varepsilon \left(G\right)$ and establish an upper bound for its $\varepsilon$-spectral radius when $G$ is a self-centered graph.

Further, we explore the structure of $\varepsilon (G\circ H)$, where $G\circ H$ is the corona product of a self-centered graph $G$ and a graph $H$. We characterize the irreducibility of $\varepsilon (G\circ H)$ and, in this process, find that it is independent of $\varepsilon \left(H\right)$, which allows us to construct infinitely many graphs with irreducible eccentricity matrix. Moreover, we compute the complete spectrum of $\varepsilon (G\circ H)$ including its $\varepsilon$-eigenvectors, $\varepsilon$-energy, and $\varepsilon$-inertia. Finally, we conclude that several non-isomorphic $\varepsilon$-co-spectral graphs can be generated using the corona product of two graphs.