Spectrum of two new joins of graphs and infinite families of integral graphs

Gopalapillai Indulal

Kragujevac Journal of Mathematics, Tome 36 (2012) no. 1, p. 133

Résumé

Let $G_1$ and $G_2$ be two graphs with $V(G_1)=\{u_1,u_2,\dots,u_{n_1}\}$, $E(G_1)=\{e_1,e_2,\dots,e_{m_1}\}$ and $V(G_2)=\{v_1,v_2,\dots,v_{n_2}\}$. Let $ S(G) $ denote the subdivision graph of $G$ [3]. The $S_{vertex}$ join of $G_1$ with $G_2$, denoted by $ G_1 \veedot 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 $S_{edge}$ join of $G_1$ with $G_2$, denoted by $ G_1 \veebar 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 the spectrum of these two new joins of graphs. As an application some infinite family of new classes of integral graphs are constructed.

Détail

Classification : 05C 05C12
Keywords: Adjacency matrix, Characteristic Polynomial, Join of graphs