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Unbalanced complete bipartite signed graphs ${K_{m, n}}^{\sigma}$ having $m$ and $n$ as Laplacian eigenvalues with maximum multiplicities | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 18، دوره 11، شماره 2، شهریور 2026، صفحه 629-642 اصل مقاله (938.79 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29897.2214 | ||
| نویسنده | ||
| Tahir Shamsher* | ||
| School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, 752050, India | ||
| چکیده | ||
| A signed graph $ {G}^{\sigma} = (G, \sigma) $ consists of an underlying graph $ G = (V, E) $ along with a signature function $ \sigma: E \rightarrow \{-1, 1\} $. A cycle in a signed graph is termed positive if it contains an even number of negative edges, and negative if it contains an odd number of negative edges. A signed graph is considered { balanced} if it has no negative cycles; otherwise, it is { unbalanced}. Let $K_{m,n}$ be a { complete bipartite graph} on $m+n$ vertices. It is well known that for a balanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $, the parameters $ m $ and $ n $ are Laplacian eigenvalues with multiplicities $ n-1 $ and $ m-1 $, respectively. This raises a natural question about the maximum multiplicities of Laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $. In this paper, we demonstrate that the multiplicities of the Laplacian eigenvalues $ m $ and $ n $ in an unbalanced complete bipartite signed graph $ {K_{m,n}}^{\sigma} $ are at most $ n-2 $ and $ m-2 $, respectively. Additionally, we characterize all the signed graphs for which $ m $ and $ n $ are Laplacian eigenvalues with these maximum multiplicities. | ||
| کلیدواژهها | ||
| signed graph؛ maximum multiplicity؛ Laplacian matrix؛ complete bipartite signed graph؛ minimum multiplicity | ||
| مراجع | ||
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