Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

The signless Laplacian polynomial of a graph is the characteristic polynomial of the matrix , where is the diagonal degree matrix and is the adjacency matrix of . In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs. 1. Introduction Let be a simple graph with vertices and edges. Let the vertex set of be . Let denote the degree of a vertex in . The adjacency matrix of a graph is , where if is adjacent to and , otherwise. The characteristic polynomial of a graph is defined as , where is an identity matrix of order . The degree matrix of a graph is the diagonal matrix , where , . The matrix , is called the Laplacian matrix and the matrix is called the signless Laplacian matrix or -matrix of . The characteristic polynomial of , defined as , is called the Laplacian polynomial of . The characteristic polynomial of , defined as , is called the signless Laplacian polynomial or -polynomial of a graph , where is an identity matrix of order . Several results on Laplacian of are reported in the literature (see, e.g., [1–4]). Recently signless Laplacian attracted the attention of researchers [5–12]. In [13], the Laplacian polynomial of a graph is expressed in terms of the characteristic polynomial of the induced subgraphs. In this paper we express signless Laplacian polynomial of a graph in terms of the characteristic polynomial of its induced subgraphs. Further the signless Laplacian polynomial of a regular graph is expressed in terms of the derivatives of its characteristic polynomial. Using these results, we express characteristic polynomial of line graph and of subdivision graph in terms of the characteristic polynomial of its induced subgraphs. We use standard terminology of graph theory [14]. 2. Signless Laplacian Polynomial in terms of Characteristic Polynomial Let the set , . Note that . We denote the product of degrees of the vertices of which belongs to by , that is, . The graph is an induced subgraph of with vertex set . If , then , a graph without vertices. Note that . Theorem 1 (see [15] ( derivative)). Let be a graph with vertices, then Theorem 2. Let be a graph with vertices, then Proof. Let , be the adjacency matrix of and , where , . Then, Splitting the determinant of (3) as a sum of two