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`Journal of Discrete MathematicsVolume 2013 (2013), Article ID 105624, 4 pageshttp://dx.doi.org/10.1155/2013/105624`
Research Article

Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

1Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum 590008, India
2Department of Mathematics, B.V.Bhoomaraddi College of Engineering & Technology, Hubli 580031, India

Received 21 July 2012; Accepted 6 September 2012

Copyright © 2013 Harishchandra S. Ramane et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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., [14]). Recently signless Laplacian attracted the attention of researchers [512]. 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 determinants, we get
Again splitting each of the determinants of (4) as a sum of two determinants and continuing the procedure in succession, at step we get
(in the above expression, stands for the identity matrix of the appropriate order and is a vertex deleted subgraph of ) and

Corollary 3. If is an -regular graph with vertices, then

Proof. The graph is an -regular graph, therefore for all . Substituting this in (2) gives that

3. Characteristic Polynomial of Line Graph and Subdivision Graph

The line graph of a graph is the graph whose vertex set has one-to-one correspondence to the edge set of and two vertices in are adjacent if and only if the corresponding edges have a vertex in common in .

Theorem 4. Let be a graph with vertices and edges and is the line graph of , then

Proof. If is a graph with vertices and edges, then [8]
From (2),
Substituting (11) in (10), the result follows.

Corollary 5. If is an -regular graph with vertices, then

Proof. The graph is an -regular graph, therefore for all and the number of edges of is .
Therefore, from (9),

The subdivision graph of a graph is the graph obtained from by inserting a new vertex into every edge of .

Theorem 6. Let be a graph with vertices, edges and is the subdivision graph of , then

Proof. If is a graph with vertices and edges, then [8]
From (2),
Substituting (16) in (15), the result follows.

Corollary 7. If is an -regular graph with vertices, then

Proof. As is an -regular graph, for all and the number of edges of is .
Therefore, from (14),

4. Number of Spanning Trees

If is an -regular graph, then

which gives that

Also for an -regular graph [3, 16],

Let be the number of spanning trees of , then [3, 16]

Here we give the number of spanning trees using signless Laplacian.

Theorem 8. If is an -regular graph with vertices, then

Proof. From (22),

5. Conclusion

Equation (2) establishes the relationship between the signless Laplacian polynomial and characteristic polynomial of a graph. Equations (9) and (14) express the characteristic polynomial of line graph and of subdivision graph in terms of the characteristic polynomial of a graph. As corollaries of these results, (12) and (17) give the characteristic polynomial of line graph and the subdivision graph of a regular graph in terms of the derivatives of the characteristic polynomial of a graph. These results are different from those obtained in [17, 18].

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