## Combinatorics and Graph Theory   ## Spectra of a New Join in Duplication Graph

Authors: K. Reji Kumar, Renny P. Varghese

The Duplication graph DG of a graph G, is obtained by inserting new vertices corresponding to each vertex of G and making the vertex adja- cent to the neighbourhood of the corresponding vertex of G and deleting the edges of G. Let G1 and G2 be two graph with vertex sets V (G1) and V (G2) respectively. The DG - vertex join of G1 and G2 is denoted by G1 t G2 and it is the graph obtained from DG1 and G2 by joining every vertex of V (G1) to every vertex of V (G2). The DG - add vertex join of G1 and G2 is denoted by G1 ./ G2 and is the graph obtained from DG1 and G2 by joining every additional vertex of DG1 to every vertex of V (G2). In this paper we determine the A - spectra and L - spectra of the two new joins of graphs for a regular graph G1 and an arbitrary graph G2 . As an application we give the number of spanning tree, the Kirchhoff index and Laplace energy like invariant of the new join. Also we obtain some infinite family of new class of integral graphs