The Fourth and Fifth Laplacian Coefficients of some Rooted Trees

Document Type: Original Scientific Paper


1 Department of Mathematics, Islamic Azad University, Science and Researcher Branch Tehran, I. R. Iran

2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran


Abstract. The Laplacian characteristic polynomial of an n-vertex graph G has the form f(G,x) = x^n+∑l_ix^n-i. In this paper, the fourth and fifth coefficient of f(G,x), will be investigated, where G is a T(k,t) tree in which a rooted tree with degree sequence k,k,···,k,1,1,···,1 is denoted by T(k,t).


[1] A. R. Ashrafi, M. Eliasi and A. Ghalavand, Laplacian coefficients and Zagreb
indices of trees, Linear Multilinear Algebra 67 (2019) 1736–1749.
[2] N. Biggs, Algebraic Graph Theory, Cambridge Univ Press, Cambridge, 1993.
[3] A. Behmaram, On the number of 4-matchings in graphs, MATCH Commun.
Math. Comput. Chem. 62 (2009) 381–388.
[4] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra in Graph–Theory and Application,
Academic Press, New York, 1980.
[5] G. H. Fath-Tabar, A. R. Ashrafi and I. Gutman, Note on Estrada and LEstrada
indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math. 139 (2009)
[6] G. H. Fath-Tabar, T. Došlic and A. R. Ashrafi, On the Szeged and the Laplacian
Szeged spectrum of a graph, Linear Algebra Appl. 433 (2010) 662–671.
[7] G. H. Fath-Tabar and A. R. Ashrafi, Some remarks on Laplacian eigenvalues
and Laplacian energy of graphs, Math. Commun. 15 (2010) 443–451.

[8] E. J. Farrel, J. M. Guo and G. M. Constantine, On matching coefficients,
Discrete Math. 89 (1991) 203–210.
[9] I. Gutman and L. Pavlovic, On the coefficients of the Laplacian characteristic
polynomial of trees, Bull. Cl. Sci. Math. Nat. Sci. Math. 28 (2003) 31–40.
[10] C. S. Oliveira, N. M. Maia de Abreu and S. Jurkiewicz, The characteristic
polynomial of the Laplacian of graphs in (a; b)-linear classes, Special issue on
algebraic graph theory, Linear Algebra Appl. 356 (2002) 113–121.
[11] F. Taghvaee and G. H. Fath-Tabar, On the skew spectral moments of graphs,
Trans. Comb. 6 (2017) 47–54.
[12] F. Taghvaee and G. H. Fath-Tabar, Relationship between coefficients of characteristic
polynomial and matching polynomial of regular graphs and its applications,
Iranian J. Math. Chem. 8 (2017) 7–23.

Volume 4, Issue 2
Special Issue: Spectral Graph Theory and Mathematical Chemistry with Connection to Computer Science
Autumn 2019
Pages 183-192