On Laplacian resolvent energy of graphs

Document Type : Research Paper


1 Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India

2 Department of Mathematics, University of Kashmir, Srinagar, India


Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}>\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.


Main Subjects

[1] L. E. Allem, J. Capaverde, V. Trevisan, I. Gutman, E. Zogic and E. Glogic, Resolvent energy of unicyclic, bicyclic
and tricyclic graphs, MATCH Commun. Math. Comput. Chem., 77 (2017) 95–104.
[2] A. Cafure, D. A. Jaume, L. N. Grippo, A. Pastine, M. D. Safe, V. Trevisan and I. Gutman, Some results for the
(signless) Laplacian resolvent, MATCH Commun. Math.Comput. Chem., 77 (2017) 105–114.
[3] Z. Du, Asymptotic expressions for resolvent energies of paths and cycles, MATCH Commun. Math. Comput. Chem.,
77 (2017) 85–94.
[4] A. Farrugia, The increase in the resolvent energy of a graph due to the addition of a new edge, Appl. Math. Comput.,
321 (2018) 25–36.
[5] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem., 53 (2015) 1184–1190.
[16] P. Milosevic, E. Milovanovic, M. Matejic and I. Milovanovic, On relations between Kirchhoff index, Laplacian energy,
Laplacian-energy-like invariant and degree deviation of graphs, Filomat, 34 (2020) 1025–1033.
[17] I. Milovanovic, M. Matejic, E. Glogic and E. Milovanovic, Some new lower bounds for the Kirchhoff index of a
graph, Bull. Aust. Math. Soc., 97 (2018) 1–10.
[18] S. Pirzada and H. A. Ganie, On Laplacian-energy-like invariant and Incidence energy, Trans. Comb., 4 (2015) 25–35.
[19] S. Pirzada, H. A.Ganie and I. Gutman, On on Laplacian-energy-like invariant and Kirchhoff index, MATCH Com-
mun. Math. Comput. Chem., 73 (2015) 41–60.
[20] S. Pirzada, H. A. Ganie and I. Gutman, Comparison between Laplacian-energy-like invariant and the Kirchhoff
index, El. J. Linear Algebra, 31 (2016) 27–41.
[21] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlackSwan Hyderabad, 2012.
[22] J. Radon, Theorie und Anwendungen der absolut Additiven Mengenfunktionnem, Sitzungsber Acad. Wissen, Wien,
122 (1913) 1295–1438.
[23] B. C. Rennie, On a class of inequalities, J. Australian Math. Soc., 3 (1963) 442–448.
[24] H. Wiener, Structural determination of paraffin boilling points, J. Amer. Chem. Soc., 69 (1947) 17–20.
[25] Z. Zhu, Some extremal properties of the resolvent energy, Estrada and resolvent Estrada indices of graphs, J. Math.
Anal. Appl., 447 (2017) 957–970.
[6] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl., 414 (2006) 29–37.
[7] I. Gutman and B. Mohar, The quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci., 36
(1996) 982–985.
[8] I. Gutman, B. Furtula, E. Zogic and E. Glogic, Resolvent energy of graphs, MATCH Commun. Math. Comput.
Chem., 75 (2016) 279–290.
[9] I. Gutman, B. Furtula, E. Zogic and E. Glogic, Resolvent energy of graphs, MATCH Commun. Math. Comput.
Chem., 75 (2016) 279–290.
[10] D. J. Klein and M. Randic, Resistence distance, J. Math. Chem., 12 (1993) 81–95.
[11] J. Liu and B. A. Liu, Laplacian-energy-like invariant, MATCH Commun. Math. Comput. Chem., 59 (2008) 355–372.
[12] B. Liu and Z. Y. Liu, A survey on the Laplacian-energy-like invariant, MATCH Commun. Math. Comput. Chem.,
66 (2011) 713–730.
[13] M. Matejic, E. Zogic, E. Milovanovic and I. Milovanovic, A note on the Laplacian resolvent energy of graphs, Asian
European J. Math., 13 (2020) 6 p.
[14] I. Milovanovic, E. Milovanovic, E. Glogic and M. Matejic, On Kirchhoff index, Laplacian energy and their relation,
MATCH Commun. Math. Comput. Chem., 81 (2019) 405–418.
[15] E. I. Milovanovic, I. Z. Milovanovic and M. M. Matejic, On relation between Kirchhoff index and Laplacian-energy-
like invariant of graphs, Math. Int. Res., 2 (2017) 141–154.