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BY 4.0 license Open Access Published by De Gruyter Open Access December 13, 2019

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Mohammad Adm, Shaun Fallat, Karen Meagher, Shahla Nasserasr, Sarah Plosker and Boting Yang
From the journal Special Matrices

Abstract

Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition.

We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

MSC 2010: 05C50; 15A18

References

[1] J. Ahn, C. Alar, B. Bjorkman, S. Butler, J. Carlson, A. Goodnight, H. Knox, C. Monroek, and M.C. Wigal, Ordered multiplicity inverse eigenvalue problem for graphs on six vertices, arXiv preprint arXiv:1708.02438 2017.Search in Google Scholar

[2] B. Ahmadi, F. Alinaghipour, M.S. Cavers, S. Fallat, K. Meagher, and S. Nasserasr, Minimum number of distinct eigenvalues of graphs, Electronic Journal of Linear Algebra, 26 (2013) pp. 673–691.Search in Google Scholar

[3] W. Barrett, S. Butler, S.M. Fallat, H.T. Hall, L. Hogben, J.C.-H. Lin, B.L. Shader, and M. Young, The inverse eigenvalue problem of a graph: Multiplicities and minors, Journal of Combinatorial Theory, Series B (2019), https://doi.org/10.1016/j.jctb.2019.10.005.10.1016/j.jctb.2019.10.005Search in Google Scholar

[4] W. Barrett, S. Fallat, H.T. Hall, L. Hogben, J.C.-H. Lin, and B.L. Shader, Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph, Electronic Journal of Combinatorics 24 (2) (2017) pp. 2–40.Search in Google Scholar

[5] W. Barrett, S. Fallat, H.T. Hall, L. Hogben, J.C.-H. Lin, and B.L. Shader, Low values of q(G). To be submitted.Search in Google Scholar

[6] W. Barrett, H.T. Hall, and H. van der Holst The inertia set of the join of graphs, Linear Algebra and its Applications, 434 (2011) pp. 2197–2203.Search in Google Scholar

[7] W. Barrett, H. van der Holst, and R. Loewy, Graphs whose minimal rank is two, Electronic Journal of Linear Algebra, 11 (2004) pp. 258–280.Search in Google Scholar

[8] B. Bjorkman, L. Hogben, S. Ponce, C. Reinhart, and T. Tranel, Applications of analysis to the determination of the minimum number of distinct eigenvalues of a graph, Pure and Applied Functional Analysis, 3 (2018) pp. 537–563.Search in Google Scholar

[9] S.M. Fallat and L. Hogben, The minimum rank of symmetric matrices described by a graph: a survey, Linear Algebra and its Applications, 426 (2007) pp. 558–582.Search in Google Scholar

[10] I.-J. Kim and B.L. Shader, Smith normal form and acyclic matrices, Journal of Algebraic Combinatorics, 29 (2009) pp. 63–80.Search in Google Scholar

[11] R.H. Levene, P. Oblak, and H. Šmigoc, A Nordhaus-Gaddum conjecture for the minimum number of distinct eigenvalues of a graph, Linear Algebra and its Applications, 564 (2019), pp. 236–263.Search in Google Scholar

[12] Z. Chen, M. Grimm, P. McMichael, and C.R. Johnson, Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues, Linear Algebra and its Applications, 458 (2014) pp. 403–428.Search in Google Scholar

[13] K. Meagher and I. Sciriha, Graphs that have a weighted adjacency matrix with spectrum {λ1n−2,λ12}, arXiv preprint arXiv:1504.04178, 2015.Search in Google Scholar

[14] K.H. Monfared and B.L. Shader, The nowhere-zero eigenbasis problem for a graph, Linear Algebra and its Applications, 458 (2016) pp. 296–312.Search in Google Scholar

[15] P. Oblak and H. Šmigoc, Graphs that allow all the eigenvalue multiplicities to be even, Linear Algebra and its Applications, 454 (2014) pp. 72–90.Search in Google Scholar

[16] P. Oblak and H. Šmigoc, The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph, Linear Algebra and its Applications, 512 (2017) pp. 48–70.Search in Google Scholar

[17] T. Peters, Positive semidefinite maximum nullity and zero forcing number, Electronic Journal of Linear Algebra, 23 (2012) pp. 815–830.Search in Google Scholar

Received: 2019-08-01
Accepted: 2019-12-02
Published Online: 2019-12-13

© 2019 Mohammad Adm et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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