On Path Laplacian Eigenvalues and Path Laplacian Energy of Graphs

Year 2018, Issue: 20 , 93 - 101 , 28.01.2018

Shridhar Chandrakant Patekar Maruti Mukinda Shikare

https://izlik.org/JA48JT93DB

Abstract

We introduce the concept of Path Laplacian Matrix for a graph and explore the eigenvalues of this matrix. The eigenvalues of this matrix are called the path Laplacian eigenvalues of the graph. We investigate path Laplacian eigenvalues of some classes of graph. Several results concerning path Laplacian eigenvalues of graphs have been obtained.

Keywords

Path , Real symmetric matrix , Laplacian matrix

References

• [1] Lowel W. Beineke, Robin J. Wilson, Topics in Algebraic Graph Theory, Cambridge University Press, 2004.
• [2] Douglas B.West, Introduction to Graph theory, Prentice-Hall, U.S.A, 2001.
• [3] R. B. Bapat, Graphs and Matrices, Hindustan Book agency, New Delhi, 2010.
• [4] Brouwer A. E., Haemers W. E., Spectra of Graphs, Springer, New York, 2010.
• [5] Varga, R. S., Matrix Iterative Analysis, Springer-Verlag, Berlin, 2000.
• [6] S. C. Patekar, M. M. Shikare, On the Path Matrices of Graphs and Their Properties, Advances and Applications in Discrete Mathematics, Vol. 17. N0. 2, (2016), pp 169- 184.
• [7] M. M. Shikare, P. P. Malavadkar, S. C. Patekar, I. Gutman, On Path Eigenvalues and Path Energy of Graphs, MATCH Communications in Mathematical and in Computer Chemistry, Vol. 79. N0. 2, (2018), pp 387-398.
• [8] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221-229.
• [9] R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218-238.
• [10] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. (1994) 143-176.
• [11] R. Merris, A survey of graph Laplacians, Linear Multilinear Algebra 39 (1995) 19-31.