On the Nullity of Expanded Graphs

The nullity (degree of singularity) η(G) of a graph G is the multiplicity of zero as an eigenvalue in its spectrum. It is proved that, the nullity of a graph is the number of non-zero independent variables in any of its high zero-sum weightings. Let u and v be nonadjacent coneighbor vertices of a connected graph G, then η(G) = η(G−u) + 1 = η(G−v) + 1. If G is a graph with a pendant vertex (a vertex with degree one), and if H is the subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then η(G) = η(H). Let H be a graph of order n and G1, G2,…, Gn be given vertex disjoint graphs, then the expanded graph is a graph obtained from the graph H by replacing each vertex vi of H by a graph Gi with extra sets of edges Si,j for each edge vivj of H in which Si,j = {uw: u∈V(Gi), w∈V(Gj)}. In this research, we evaluate the nullity of expanded graphs, for some special ones, such as null graphs, complete bipartite graphs, star graphs, complete graphs, nut graphs, paths, and cycles.