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Maximizing the spectral radius of bicyclic graphs with fixed girth. (English) Zbl 1227.05189

Summary: Let \(\mathcal B (n,g)\) be the set of bicyclic graphs on \(n\) vertices with girth \(g\). In this paper, we determine the unique graph with the maximal spectral radius among all graphs in \(\mathcal B (n,g)\). Moreover, the maximal spectral radius is a decreasing function on \(g\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C38 Paths and cycles
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References:

[1] Berman, A.; Zhang, X., On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B, 83, 233-240 (2001) · Zbl 1023.05098
[2] Brualdi, R. A.; Solheid, E. S., On the spectral radius of connected graphs, Publ. Inst. Math. (Beograd), 39, 53, 45-54 (1986) · Zbl 0603.05028
[3] Guo, S., On the spectral radius of bicyclic graphs with \(n\) vertices and diameter \(d\), Linear Algebra Appl., 422, 119-132 (2007) · Zbl 1112.05064
[4] Hansen, P.; Stevanović, D., On bags and bugs, Discrete Appl. Math., 156, 986-997 (2008) · Zbl 1145.05028
[5] Hoffman, A. J.; Smith, J. H., (Fiedler, Recent Advances in Graph Theory (1975), Academia: Academia Praha, New York), 273-281
[6] Li, Q.; Feng, K., On the largest eigenvalues of graphs, Acta Math. Appl. Sinica, 2, 167-175 (1979), (in Chinese)
[7] Schwenk, A. J., Computing the characteristic polynomial of a graph, (Graphs and Combinatorics. Graphs and Combinatorics, Lectures Notes in Mathematics, vol. 406 (1974), Springer: Springer Berlin), 153-172
[8] Van Dam, E. R., Graphs with given diameter maximizing the spectral radius, Linear Algebra Appl., 426, 454-457 (2007) · Zbl 1122.05063
[9] B. Wu, Graft transformation of graphs and spectral radius, Master Thesis, East China Normal University, 2003.; B. Wu, Graft transformation of graphs and spectral radius, Master Thesis, East China Normal University, 2003.
[10] Yu, A.; Tian, F., On the spectral radius of bicyclic graphs, Match Commun. Math. Comput. Chem., 52, 91-101 (2004) · Zbl 1080.05522
[11] Zhai, M.; Liu, R.; Shu, J., On the spectral radius of bipartite graphs with given diameter, Linear Algebra Appl., 430, 1165-1170 (2009) · Zbl 1226.05206
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