On co-regular signed graphs

A graph whose edges are labeled either as positive or negative is called a signed graph. A signed graph is said to be net-regular if every vertex has constant net-degree k, namely, the difference between the number of positive and negative edges incident with a vertex. In this paper, we analyze some properties of co-regular signed graphs which are net-regular signed graphs with the underlying graphs being regular. An r-regular graph G is said to be co-regularizable with the co-regularity pair (r, k) if we can find a signed graph Σ of net-degree k with the underlying graph G. Net(G) is the set of all possible values of net-degree k, such that G is co-regularizable with the co-regularity pair (r, k); we find Net(G) for some special classes of graphs. Also we establish a connection between co-regularizability of G and its graph factors. An algorithm for producing a co-regular signed Harary graph, of which the co-regular signed graph on the complete graph Kn is a particular example, is included. We also establish the fact that every signed graph can be embedded as an induced subgraph of a net-regular or co-regular signed graph. S.K. HAMEED ET AL. /AUSTRALAS. J. COMBIN. 62 (1) (2015), 8–17 9