Remarks on the upper bound for the Randić energy of bipartite graphs. (English) Zbl 1357.05080
Summary: Let \(G = (V, E)\), \(V = \{1, 2, \ldots, n \}\) be a simple graph without isolated vertices, with \(n\)\((n \geq 3)\) vertices and \(m\) edges, whose vertex degrees are given in the following form \(d_1 \geq d_2 \geq \cdots \geq d_n > 0\). If \(A\) is the adjacency matrix, the Randić matrix \(R = \| R_{i j} \|\) is defined in the following way
\[ R_{i j} = \begin{cases} \frac{1}{\sqrt{d_i d_j}} & \text{if } v_i \text{ and } v_j \text{ are adjacent,} \\ 0 & \text{otherwise .} \end{cases} \]
The eigenvalues of matrix \(R\), \(\rho_1 \geq \rho_2 \geq \cdots \geq \rho_n\), are called the Randić eigenvalues of graph \(G\). The Randić energy of graph \(G\), denoted by \(R E\), is defined in the following way: \(R E = R E(G) = \sum_{i = 1}^n | \rho_i | \text{.}\) In this paper, upper bounds for graph invariant RE have been studied.
\[ R_{i j} = \begin{cases} \frac{1}{\sqrt{d_i d_j}} & \text{if } v_i \text{ and } v_j \text{ are adjacent,} \\ 0 & \text{otherwise .} \end{cases} \]
The eigenvalues of matrix \(R\), \(\rho_1 \geq \rho_2 \geq \cdots \geq \rho_n\), are called the Randić eigenvalues of graph \(G\). The Randić energy of graph \(G\), denoted by \(R E\), is defined in the following way: \(R E = R E(G) = \sum_{i = 1}^n | \rho_i | \text{.}\) In this paper, upper bounds for graph invariant RE have been studied.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
References:
| [1] | Alikhani, S.; Ghanbari, N., Randić energy of specific graphs, Appl. Math. Comput., 269, 722-730 (2015) · Zbl 1352.05108 |
| [2] | Bozkurt, Ş. B.; Bozkurt, D., Sharp Upper Bounds for Energy and Randić Energy, MATCH Commun. Math. Comput. Chem., 70, 669-680 (2013) · Zbl 1299.05207 |
| [3] | Bozkurt, Ş. B.; Güngör, A. D.; Gutman, I., Randić spectral Radius and Randić energy, MATCH Commun. Math. Comput. Chem., 64, 321-334 (2010) · Zbl 1265.05351 |
| [4] | Cauers, M., The normalized Laplacian matrix and general Randić index of graphs (2010), University of Regina, (Ph.D. thesis) |
| [5] | Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs-Theory and Application (1980), Academic Press: Academic Press New York · Zbl 0458.05042 |
| [6] | Das, K. C.; Sorgun, S., On Randić energy of graphs, MATCH Commun. Math. Comput. Chem., 72, 227-238 (2014) · Zbl 1464.05232 |
| [7] | Das, K. C.; Sorgun, S.; Gutman, I., On Randić energy, MATCH Commun. Math. Comput. Chem., 73, 81-92 (2015) · Zbl 1464.05233 |
| [8] | Das, K. C.; Sorgun, S.; Xu, K., On Randić Energy of Graphs, (Graph Energies - Theory and Applications (2016), Univ. Kragujevac: Univ. Kragujevac Kragujevac), 111-122 |
| [9] | Gu, R.; Li, X.; Liu, J., Note on three results on Randić energy and incidence energy, MATCH Commun. Math. Comput. Chem., 73, 61-71 (2015) · Zbl 1464.05237 |
| [10] | Gutman, I., The energy of a graph: Old and new result, (Algebraic Combinatorics and Applications (2001), Springer-Verlag: Springer-Verlag Berlin), 196-211 · Zbl 0974.05054 |
| [11] | Gutman, I., Topology and stability of conjugated hydrocarbons. the dependence of total \(\pi \)-electron energy on molecular topology, J. Serb. Chem. Soc., 70, 441-456 (2005) |
| [12] | Gutman, I.; Furtula, B.; Bozkurt, Ş., On Randić energy, Linear Algebra Appl., 442, 50-57 (2014) · Zbl 1282.05118 |
| [13] | Gutman, I.; Li, X., Graph Energies - Theory and Applications (2016), Univ. Kragujevac: Univ. Kragujevac Kragujevac |
| [14] | Gutman, I.; Li, X.; Zhang, J., Graph energy, (Dehmer, M.; Emmert-Streib, F., Analysis of Complex Networks. From Biology to Linguistics (2009), Wiley-VCH: Wiley-VCH Weinheim), 145-174 · Zbl 1256.05140 |
| [15] | Li, J.; Guo, J. M.; Shiu, W. C., A note on Randić energy, MATCH Commun. Math. Comput. Chem., 74, 389-398 (2015) · Zbl 1462.05089 |
| [16] | Li, X.; Shi, Y.; Gutman, I., Graph Energy (2012), Springer: Springer New York · Zbl 1262.05100 |
| [17] | Li, X.; Wang, J., Randić energy and Randić eigenvalues, MATCH Commun. Math. Comput. Chem., 73, 73-80 (2015) · Zbl 1464.05242 |
| [18] | Li, X.; Yang, Y., Sharp Bounds for the General Randić Index, MATCH Commun. Math. Comput. Chem., 51, 155-166 (2004) · Zbl 1053.05067 |
| [19] | Liu, B. L.; Huang, Y. F.; Feng, J. F., A Note on the Randić Spectral Radius, MATCH Commun. Math. Comput. Chem., 68, 913-916 (2012) · Zbl 1289.05133 |
| [21] | Maden, A. D., New bounds on the incidence energy, Randić energy and Randić Estrada index, MATCH Commun. Math. Comput. Chem., 74, 367-387 (2015) · Zbl 1462.05093 |
| [22] | Majstorović, S.; Klobučar, A.; Gutman, I., Selected topics from the theory of graph energy: hypoenergetic graphs, (Applications of Graph Spectra (2009), Math. Inst: Math. Inst Belgrade), 65-105 · Zbl 1224.05313 |
| [23] | Milovanović, I.; Milovanović, E.; Gutman, I., Upper bounds for some graph energies, Appl. Math. Comput., 289, 435-443 (2016) · Zbl 1410.05138 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
