Energy, Laplacian energy of double graphs and new families of equienergetic graphs

Abstract

For a graph $G$ with vertex set $V(G)=\{v_1,v_2,\cdots ,v_n\}$, the extended double cover $G^{\ast }$ is a bipartite graph with bipartition (X, Y), $X=\{x_1,x_2,\cdots ,x_n\}$ and $Y=\{y_1,y_2,\cdots ,y_n\}$, where two vertices $x_i$ and $y_j$ are adjacent if and only if $i=j$ or $v_i$ adjacent to $v_j$ in $G$. The double graph $D[G]$ of $G$ is a graph obtained by taking two copies of $G$ and joining each vertex in one copy with the neighbours of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs $G^{\ast }$ and $D[G]$, $L$-spectra of $G^{k\ast }$ the $k$-th iterated extended double cover of $G$. We obtain a formula for the number of spanning trees of $G^{\ast }$. We also obtain some new families of equienergetic and $L$-equienergetic graphs.

arXiv comment23 pages, 1 figure