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

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph

Graphical sequences of some family of induced subgraphs

The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. The $S_{vertex}$ or $S_{ver}$ join of the graph $G_{1}$ with the graph $G_{2}$, denoted by $G_{1}\dot{\vee}G_{2}$, is obtained from $S(G_{1})$ and $G_{2}$ by joining all vertices of $G_{1}$ with all vertices of $G_{2}$. The $S_{edge}$ or $S_{ed}$ join of $G_{1}$ and $G_{2}$, denoted by $G_{1}\bar{\vee}G_{2}$, is obtained from $S(G_{1})$ and $G_{2}$ by joining all vertices of $S(G_{1})$ corresponding to the edges of $G_{1}$ with all vertices of $G_{2}$. In this paper, we obtain graphical sequences of the family of induced subgraphs of $S_{J} = G_{1}\vee G_{2}$, $S_{ver} = G_{1}\dot{\vee}G_{2}$ and $S_{ed} = G_{1}\bar{\vee}G_{2}$. Also we prove that the graphic sequence of $S_{ed}$ is potentially $K_{4}-e$-graphical

Recognition of split-graphic sequences

Using different definitions of split graphs we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular, and graphic sequences