Journal of Discrete Mathematics

Volume 2013 (2013), Article ID 105624, 4 pages

http://dx.doi.org/10.1155/2013/105624

## Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

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

Received 21 July 2012; Accepted 6 September 2012

Academic Editor: Kinkar C. Das

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.