Abstract
The multiple subdivision graph of a graph [Formula: see text], denoted by [Formula: see text], is the graph obtained by inserting [Formula: see text] paths of length 2 replacing every edge of [Formula: see text]. When [Formula: see text], [Formula: see text] is the subdivision graph of [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges, [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges. The quasi-corona SG-vertex join [Formula: see text] of [Formula: see text] and [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] copies of [Formula: see text] by joining every vertex of [Formula: see text] to every vertex of [Formula: see text], and multiple SG-vertex join [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] by joining every vertex of [Formula: see text] to every vertex of [Formula: see text]. 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 [Formula: see text] being a regular graph. Then we obtain their adjacency spectra for the case of [Formula: see text] and [Formula: see text] being regular graphs.
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