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

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \cdots, v_n\}$, the extended double cover $G^*$ 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 $G^*$ and $D[G]$, $L$-spectra of $G^{k*}$ the $k$-th iterated extended double cover of $G$. We obtain a formula for the number of spanning trees of $G^*$. We also obtain some new families of equienergetic and $L$-equienergetic graphs.Comment: 23 pages, 1 figur

On the Randi\'{c} index and conditional parameters of a graph

The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right.$$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.Comment: arXiv admin note: text overlap with arXiv:math/060243