Spectra of some new extended corona

Abstract

For two graphs $mathrm{G}$ and $mathrm{H}$ with $n$ and $m$ vertices, the corona $mathrm{G}circmathrm{H}$ of $mathrm{G}$ and $mathrm{H}$ is the graph obtained by taking one copy of $mathrm{G}$ and $n$ copies of $mathrm{H}$ and then joining the $i^{th}$ vertex of $mathrm{G}$ to every vertex in the $i^{th}$ copy of $mathrm{H}$. The neighborhood corona $mathrm{G}starmathrm{H}$ of $mathrm{G}$ and $mathrm{H}$ is the graph obtained by taking one copy of $mathrm{G}$ and $n$ copies of $mathrm{H}$ and joining every neighbor of the $i^{th}$ vertex of $mathrm{G}$ to every vertex in the $i^{th}$ copy of $mathrm{H}$. In this paper, we define four new extensions of corona and neighborhood corona of two graphs $mathrm{G}$ and $mathrm{H}$; named the identity-extended corona, identity-extended neighborhood corona, neighborhood extended corona and neighborhood extended neighborhood corona and then determine the spectrum of their adjacency matrix, where $mathrm{H}$ is a regular graph. As an application, we exhibit infinite families of integral graphs.