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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 105624, 4 pages
http://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

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.

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].

References

  1. R. Li, “Some lower bounds for Laplacian energy of graphs,” International Journal of Contemporary Mathematical Sciences, vol. 4, no. 5, pp. 219–223, 2009. View at Zentralblatt MATH
  2. Z. Liu, “Energy, Laplacian energy and Zagreb index of line graph, middle graph and total graph,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 18, pp. 895–900, 2010. View at Zentralblatt MATH
  3. B. Mohar, “The Laplacian spectrum of graphs,” in Graph Theory, Combinatorics and Applications, Y. Alavi, G. Chartrand, O. E. Ollerman, and A. J. Schwenk, Eds., pp. 871–898, John Wiley & Sons, New York, NY, USA, 1991. View at Zentralblatt MATH
  4. B. Mohar, “Graph laplacians,” in Topics in Algebraic Graph Theory, L. W. Beineke and R. J. Wilson, Eds., pp. 113–136, Cambridge University Press, Cambridge, UK, 2004. View at Zentralblatt MATH
  5. D. Cvetković, “Signless Laplacians and line graphs,” Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences Mathématiques Naturelles/Sciences Mathematiques, vol. 131, no. 30, pp. 85–92, 2005. View at Zentralblatt MATH
  6. D. Cvetković, P. Rowlinson, and S. K. Simić, “Signless Laplacians of finite graphs,” Linear Algebra and Its Applications, vol. 423, no. 1, pp. 155–171, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  7. D. Cvetković, P. Rowlinson, S. K. Simic, and S. K. Simić, “Eigenvalue bounds for the signless Laplacian,” Publications de l'Institut Mathématique, vol. 81, no. 95, pp. 11–27, 2007. View at Zentralblatt MATH
  8. D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univeristy Press, Cambridge, UK, 2010.
  9. A. Daneshgar and H. Hajiabolhassan, “Graph homomorphisms and nodal domains,” Linear Algebra and Its Applications, vol. 418, no. 1, pp. 44–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. L. Feng and G. Yu, “On three conjectures involving the signless Laplacian spectral radius of graphs,” Publications de l'Institut Mathematique, vol. 85, no. 99, pp. 35–38, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. W. H. Haemers and E. Spence, “Enumeration of cospectral graphs,” European Journal of Combinatorics, vol. 25, no. 2, pp. 199–211, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. E. R. van Dam and W. H. Haemers, “Which graphs are determined by their spectrum?” Linear Algebra and Its Applications, vol. 373, pp. 241–272, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. H. B. Walikar and H. S. Ramane, “Laplacian polynomial and number of spanning trees in terms of characteristic polynomial of induced subgraphs,” AKCE International Journal of Graphs and Combinatorics, vol. 5, no. 1, pp. 35–48, 2008. View at Zentralblatt MATH
  14. F. Harary, Graph Theory, Narosa Publishing House, New Delhi, India, 1998.
  15. H. S. Ramane, Some topics in spectral graph theory [Ph.D. thesis], Karnatak University, Dharwad, India, 2002.
  16. D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Academic Press, New York, NY, USA, 1980.
  17. H. Sachs, “Über Teiler, Faktoren und charakteristische Polynome von Graphen,” Teil I. Wiss. Z. TH Ilmenau, vol. 13, pp. 405–412, 1967. View at Zentralblatt MATH
  18. D. Cvetković, “Spectra of graphs formed by some unary operations,” Publications De L’Institut Mathe'Matique, vol. 19, no. 33, pp. 37–41, 1975. View at Zentralblatt MATH