Eigenvalues of the generalized subdivision graph with applications to graph energy

Abstract

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], let [Formula: see text] denotes the subdivision graph of [Formula: see text] with vertex set [Formula: see text]. In [Formula: see text], replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. Then in the resulting graph, replace each vertex [Formula: see text], [Formula: see text], by [Formula: see text] vertices and join every vertex to the neighbors of [Formula: see text]. The resulting graph is denoted by [Formula: see text]. This generalizes the construction of the subdivision graph [Formula: see text] to [Formula: see text] of a graph [Formula: see text]. In this paper, we provide the complete information about the spectrum of [Formula: see text] using the spectrum of [Formula: see text]. Further, we determine the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text], when [Formula: see text] is a regular graph. Also, we find the Laplacian spectrum of [Formula: see text] using the Laplacian spectrum of [Formula: see text] when [Formula: see text]. The energy of a graph [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of [Formula: see text]. The incidence energy of a graph [Formula: see text] is defined as the sum of the square roots of the signless Laplacian eigenvalues of [Formula: see text]. Finally, as an application, we show that the energy of the graph [Formula: see text] is completely determined by the incidence energy of the graph [Formula: see text].