On Laplacian resolvent energy of graphs

Bhatnagar, Sandeep; Merajuddin, Siddiqui; Pirzada, Shariefuddin

doi:10.22108/toc.2022.133236.1983

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	On Laplacian resolvent energy of graphs
Transactions on Combinatorics
مقاله 5، دوره 12، شماره 4، اسفند 2023، صفحه 217-225 اصل مقاله (445.32 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2022.133236.1983
نویسندگان
Sandeep Bhatnagar¹؛ Siddiqui Merajuddin¹؛ Shariefuddin Pirzada^* ²
¹Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India
²Department of Mathematics, University of Kashmir, Srinagar, India
چکیده
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.
کلیدواژه‌ها
Laplacian resolvent energy؛ Laplacian energy؛ Laplacian-energy-like invariant؛ Kirchhoff index

مراجع
[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.
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On Laplacian resolvent energy of graphs