×

On the eigenvalues of general sum-connectivity Laplacian matrix. (English) Zbl 1277.05086

Summary: The connectivity index was introduced by M. Randić [J. Am. Chem. Soc. 97, No. 23, 6609–6615 (1975)] and was generalized by B. Bollobás and P. Erdős [Ars Comb. 50, 225–233 (1998; Zbl 0963.05068)]. It studies the branching property of graphs, and has been applied to studying network structures.
In this paper we focus on the general sum-connectivity index which is a variant of the connectivity index. We characterize the tight upper and lower bounds of the largest eigenvalue of the general sum-connectivity matrix, as well as its spectral diameter. We show the corresponding extremal graphs. In addition, we show that the general sum-connectivity index is determined by the eigenvalues of the general sum-connectivity Laplacian matrix.

MSC:

05C35 Extremal problems in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C40 Connectivity
05C62 Graph representations (geometric and intersection representations, etc.)
15B99 Special matrices

Citations:

Zbl 0963.05068
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balister, P., Bollobás, B., Gerke, S.: The generalised Randić index of trees. J. Graph Theory 56(4), 270–286 (2007) · Zbl 1134.05017 · doi:10.1002/jgt.20267
[2] Bollobás, B., Erdös, P.: Graphs of extremal weights. Ars Comb. 50, 225–233 (1998) · Zbl 0963.05068
[3] Cao, D., Chvàtal, V., Hoffman, A.J., Vince, A.: Variations on a theorem of Ryser. Linear Algebra Appl. 260, 215–222 (1997) · Zbl 0879.05007
[4] Delorme, C., Favaron, O., Rautenbach, D.: On the Randić index. Discrete Math. 257, 29–38 (2002) · Zbl 1009.05075 · doi:10.1016/S0012-365X(02)00256-X
[5] Estrada, E., Fox, M., Higham, D.J., Oppo, G.: Network Science: Complexity in Nature and Technology. Springer, London. ISBN:978-1-84996-395-4 · Zbl 1202.68024
[6] Fiedler, M.: A property of eigenvectors of nonegative symmetric matrices and its application to graph theory. Czechoslov. Math. J. 67(100), 619–633 (1975) · Zbl 0437.15004
[7] Ghimire, J., Mani, M., Sanguankotchakorn, T., Crespi, N.: Self-connectivity estimation for super node overlay creation in ad-hoc networks. In: 2010 IEEE 17th International Conference on Telecommunications (ICT), Doha, pp. 722–727 (2010)
[8] González-Díaz, H., González-Díaz, Y., Santana, L., Ubeira, F.M., Uriarte, E.: Proteomics, networks and connectivity indices. Proteomics 8(4), 750–778 (2008). ACM Press · doi:10.1002/pmic.200700638
[9] Gutman, I., Furtula, B. (eds.) Recent Results in the Theory of Randić Index. Mathematical Chemistry Monograph, vol. 6 (2008). Kragujevac · Zbl 1294.05002
[10] Li, X., Gutman, I.: Mathematical Aspects of Randić-Type Molecular Structure Descriptors. Mathematical Chemistry Monographs, vol. 1 (2006). Kragujevac · Zbl 1294.92032
[11] Rajan, M.A., Chandra, M.G., Reddy, L.C., Hiremath, P.: A study of connectivity index of graph relevant to ad hoc networks. Int. J. Comput. Sci. Netw. Secur. 7(11), 198–203 (2007)
[12] Randić, M.: Characterization of molecular branching. J. Am. Chem. Soc. 97(23), 6609–6615 (1975) · Zbl 0770.60091 · doi:10.1021/ja00856a001
[13] Ranjan, G., Zhang, Z.: Geometry of complex networks and structural centrality (2011). CoRR arXiv:1107.0989
[14] Riera-Fernández, P., Munteanu, C.R., Escobar, M., Prado-Prado, F., Martín-Romalde, R., Pereira, D., Villalba, K., Duardo-Sánchez, A., González-Díaz, H.: New Markov–Shannon entropy models to assess connectivity quality in complex networks: from molecular to cellular pathway, parasite-host, neural, industry, and legal-social networks. J. Theor. Biol. 293, 174–188 (2012) · Zbl 06416584 · doi:10.1016/j.jtbi.2011.10.016
[15] Rodríguez, J.A.: A spectral approach to the Randic index. Linear Algebra Appl. 400, 339–344 (2005) · Zbl 1063.05097 · doi:10.1016/j.laa.2005.01.003
[16] Zhou, B., Trinajstić, N.: On a novel connectivity index. J. Math. Chem. 46, 1252–1270 (2009) · Zbl 1197.92060 · doi:10.1007/s10910-008-9515-z
[17] Zhou, B., Trinajstić, N.: On sum-connectivity matrix and sum-connectivity energy of graphs. Acta Chim. Slov. 57, 518–523 (2010)
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.