Study on adjacent spectrum of two kinds of joins of graphs.

The multiple subdivision graph of a graph G , denoted by S n (G) , is the graph obtained by inserting n paths of length 2 replacing every edge of G. When n = 1 , S 1 (G) = S (G) is the subdivision graph of G. Let G 1 be a graph with n 1 vertices and m 1 edges, G 2 be a graph with n 2 vertices and m 2 edges. The quasi-corona SG-vertex join G 1 △ G 2 of G 1 and G 2 is the graph obtained from S (G 1) ∪ G 1 and n 1 copies of G 2 by joining every vertex of G 1 to every vertex of G 2 , and multiple SG-vertex join G 1 ⊙ G 2 is the graph obtained from S n (G 1) ∪ G 1 and G 2 by joining every vertex of G 1 to every vertex of G 2 . In this paper, we calculate analytic expression of characteristic polynomial of adjacency matrix of the above two types of joins of graphs for the case of G 1 being a regular graph. Then we obtain their adjacency spectra for the case of G 1 and G 2 being regular graphs. [ABSTRACT FROM AUTHOR]