Research Projects

Spectral graph theory studies the combinatorial properties of graphs in terms of classes of matrices associated with them. The main objective of this proposal is to study the spectral properties of the signed graphs. The relationship between the chromatic number of a graph and the spectrum of the adjacency matrices is a well-studied topic in the literature. One of the main objectives of this project is to study the relationships between the signed chromatic number of signed graphs and the spectrum of the signed graphs. Also, we propose the notions of vertex chromatic number for signed graphs and indent to study their properties. We want to explore the bound for the chromatic number established by various researchers for the signed graphs. The fractional coloring of graphs is a well-studied notion for undirected graphs, and several spectral bounds are known for the same. We want to study this notion in the signed graph setup. The smallest eigenvalue of the Laplacian of a signed graph is an excellent measure of the graph frustration, that is, the smallest number of vertices to be deleted from the graph to get a balanced graph. We want to develop a theory parallel to Fiedler's theory of algebraic connectivity for Combinatorial Laplacian. Here the role of the second smallest eigenvalue is replaced by the smallest eigenvalue of the signed Laplacian. This study will apply to social networks and related fields. Clustering is one of the fundamental problems in many applications. We would like to study the theoretical aspects of signed graph clustering. As most of the social network problems are formulated in terms of signed graphs, clustering for signed graphs will have several applications in other fields.