Eigenvalues of the generalized subdivision graph with applications to graph energy

For a graph G with vertex set V-G = {v(1), v(2), . . . , v(n)} and edge set E-G = {e(1), e(2), ... , e(m)}, let S-G denotes the subdivision graph of G with vertex set V-G boolean OR E-G. In S-G, replace each vertex v(i), i = 1, 2, ... , n, by n(1) vertices and join every vertex to the neighbors of vi. Then in the resulting graph, replace each vertex e(j), j = 1, 2, ... , m, by m(1) vertices and join every vertex to the neighbors of e(j). The resulting graph is denoted by S-G(n(1), m(1)). This generalizes the construction of the subdivision graph S(G )to S-G(n(1), m(1)) of a graph G. In this paper, we provide the complete information about the spectrum of S-G(n(1), m(1)) using the spectrum of S-G. Further, we determine the Laplacian spectrum of S-G(n(1), m(1)) using the Laplacian spectrum of G, when G is a regular graph. Also, we find the Laplacian spectrum of S-G(n(1), m(1)) using the Laplacian spectrum of S(G )when n(1) = m(1). The energy of a graph G is defined as the sum of the absolute values of the eigenvalues of G. The incidence energy of a graph G is defined as the sum of the square roots of the signless Laplacian eigenvalues of G. Finally, as an application, we show that the energy of the graph S-G(n(1), m(1)) is completely determined by the incidence energy of the graph G.