Unbalanced complete bipartite signed graphs Km,nσ having m and n as Laplacian eigenvalues with maximum multiplicities

Document Type : Original paper

Author

School of Basic Sciences, IIT Bhubaneswar, Bhubaneswar, 752050, India

Abstract

A signed graph Gσ=(G,σ) consists of an underlying graph G=(V,E) along with a signature function σ:E{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 Km,n be a { complete bipartite graph} on m+n vertices. It is well known that for a balanced complete bipartite signed graph Km,nσ, the parameters m and n are Laplacian eigenvalues with multiplicities n1 and m1, respectively. This raises a natural question about the maximum multiplicities of Laplacian eigenvalues m and n in an unbalanced complete bipartite signed graph Km,nσ. In this paper, we demonstrate that the multiplicities of the Laplacian eigenvalues m and n in an unbalanced complete bipartite signed graph Km,nσ are at most n2 and m2, respectively. Additionally, we characterize all the signed graphs for which m and n are Laplacian eigenvalues with these maximum multiplicities.

Keywords

Main Subjects


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