×

Maximizing spectral radii of uniform hypergraphs with few edges. (English) Zbl 1350.05111

Summary: In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.

MSC:

05C65 Hypergraphs
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C. Berge, Graphs and Hypergraphs (North-Holland, New York-Amsterdam-Oxford, 1976).; · Zbl 0391.05028
[2] A. Bretto, Hypergraph Theory: An Introduction (Springer, Chambridge-Heidelberg- New York-Dordrecht-London, 2013).; · Zbl 1269.05082
[3] R.A. Brualdi and A.J. Hoffman, On the spectral radius of (0, 1)-matrices, Linear Algebra Appl. 65 (1985) 133-146. doi:; · Zbl 0563.15012 · doi:10.1016/0024-3795(85)90092-8
[4] R.A. Brualdi and E.S. Solheid, On the spectral radius of connected graphs, Publ. Inst. Math. (Beogard) (N.S.) 39 (1986) 45-54.; · Zbl 0603.05028
[5] K.C. Chang, K. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6 (2008) 507-520. doi:; · Zbl 1147.15006 · doi:10.4310/CMS.2008.v6.n2.a12
[6] K.C. Chang, K. Pearson and T. Zhang, On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl. 350 (2009) 416-422. doi:; · Zbl 1157.15006 · doi:10.1016/j.jmaa.2008.09.067
[7] L. Collatz and U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg 21 (1957) 63-77. doi:; · Zbl 0077.36704 · doi:10.1007/BF02941924
[8] J. Cooper and A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268-3292. doi:; · Zbl 1238.05183 · doi:10.1016/j.laa.2011.11.018
[9] S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013) 738-749. doi:; · Zbl 1261.15039 · doi:10.1016/j.laa.2011.02.042
[10] Y. Hong, On the spectra of unicyclic graph, J. East China Norm. Univ. Natur. Sci. Ed. 1 (1986) 31-34.; · Zbl 0603.05030
[11] S. Hu, L. Qi and J.-Y. Shao, Cored hypergraphs, power hypergraphs and their Lapla- cian H-eigenvalues, Linear Algebra Appl. 439 (2013) 2980-2998. doi:; · Zbl 1282.05171 · doi:10.1016/j.laa.2013.08.028
[12] H. Li, J.-Y. Shao and L. Qi,The extremal spectral radii of k-uniform supertrees, J. Comb. Optim. (2015), in press. doi:; · Zbl 1378.90084 · doi:10.1007/s10878-015-9896-4
[13] R. Merris, Degree maximal graphs are Laplacian integral , Linear Algebra Appl. 199 (1994) 381-389. doi:; · Zbl 0795.05091 · doi:10.1016/0024-3795(94)90361-1
[14] D.D. Olesky, A. Roy and P. van den Driessche, Maximal graphs and graphs with maximal spectral radius, Linear Algebra Appl. 346 (2002) 109-130. doi:; · Zbl 1002.05048 · doi:10.1016/S0024-3795(01)00504-3
[15] K. Pearson and T. Zhang, On spectral hypergraph theory of the adjacency tensor , Graphs Combin. 30 (2014) 1233-1248. doi:; · Zbl 1298.05206 · doi:10.1007/s00373-013-1340-x
[16] L. Qi, Eigenvalues of a real supersymmetric tensor , J. Symbolic Comput. 40 (2005) 1302-1324. doi:; · Zbl 1125.15014 · doi:10.1016/j.jsc.2005.05.007
[17] P. Rowlinson, On the maximal index of graphs with a prescribed number of edges, Linear Algebra Appl. 110 (1988) 43-53. doi:; · Zbl 0666.05043 · doi:10.1016/0024-3795(83)90131-3
[18] Y. Yang and Q. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl. 31 (2010) 2517-2530. doi:; · Zbl 1227.15014 · doi:10.1137/090778766
[19] J. Zhou, L. Sun, W. Wang and C. Bu, Some spectral properties of uniform hyper- graphs, Electron. J. Combin. 21 (2014) #P4.24.;
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.