Maximizing the indices of a class of signed complete graphs

Document Type : Original paper

Authors

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

The index of a signed graph is the largest eigenvalue of its adjacency matrix. Let $\mathfrak{U}_{n,k,4}$ be the set of all signed complete graphs of order $n$ whose negative edges induce a unicyclic graph of order $k$ and girth at least $4$. In this paper, we identify the signed graphs achieving the maximum index in the class $\mathfrak{U}_{n,k,4}$.

Keywords

Main Subjects

References

[1] S. Akbari, F. Belardo, F. Heydari, M. Maghasedi, and M. Souri, On the largest eigenvalue of signed unicyclic graphs, Linear Algebra Appl. 581 (2019), 145–162. https://doi.org/10.1016/j.laa.2019.06.016
[2] S. Akbari, S. Dalvandi, F. Heydari, and M. Maghasedi, Signed complete graphs with maximum index., Discuss. Math. Graph Theory 40 (2020), no. 2, 393–403. http://dx.doi.org/10.7151/dmgt.2276
[3] F. Belardo, M. Brunetti, M. Cavaleri, and A. Donno, Constructing cospectral signed graphs, Linear Multilinear Algebra 69 (2021), no. 14, 2717–2732. https://doi.org/10.1080/03081087.2019.1694483
[4] M. Brunetti and A. Ciampella, Signed bicyclic graphs with minimal index, Commun. Comb. Optim. 8 (2023), no. 1, 207–241. https://doi.org/10.22049/cco.2022.27346.1241
[5] M. Brunetti and Z. Stanić, Ordering signed graphs with large index, Ars Math. Contemp. 22 (2022), no. 4, #P4.05. https://doi.org/10.26493/1855-3974.2714.9b3
[6] M. Brunetti and Z. Stanić, Unbalanced signed graphs with extremal spectral radius or index, Comput. Appl. Math. 41 (2022), no. 3, 1–13. https://doi.org/10.1007/s40314-022-01814-5
[7] D.M. Cvetković, P. Rowlinson, and S. Simi´c, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010.
[8] N. Kafai, F. Heydari, N. Jafari Rad, and M. Maghasedi, On the signed complete graphs with maximum index, Iran. J. Sci. Technol. Trans. A Sci. 45 (2021), no. 6, 2085–2090. https://doi.org/10.1007/s40995-021-01199-w
[9] T. Koledin and Z. Stanić, Connected signed graphs of fixed order, size, and number of negative edges with maximal index, Linear Multilinear Algebra 65 (2017), no. 11, 2187–2198. https://doi.org/10.1080/03081087.2016.1265480
[10] T.V. Shijin, P. Soorya, K. Shahul Hameed, and K.A. Germina, On the distance spectra of product of signed graphs, Commun. Comb. Optim. 8 (2023), no. 1, 67–76. https://doi.org/10.22049/cco.2021.27304.1227
[11] Z. Stanić, Integral regular net-balanced signed graphs with vertex degree at most four., Ars Math. Contemp. 17 (2019), no. 1, 103–114. https://doi.org/10.26493/1855-3974.1740.803
[12] T. Zaslavsky, Matrices in the theory of signed simple graphs, Advances in Discrete Mathematics and Applications, Ramanujan Mathematical Society Lecture Notes Series 13, Ramanujan Mathematical Society, Mysore, 2010, pp. 207–229.
Communications in Combinatorics and Optimization
Volume 9, Issue 1
March 2024
Pages 169-175

Files

Share

How to cite

Statistics