Partial Chain Graphs

Chain graphs and threshold graphs are often referred to as extremal graphs, in the context that, they have the largest spectral radius among all the bipartite graphs (former one) and all the connected graphs (latter one). Nesting in the neighborhood of vertices in the above said extremal graphs have gained the attention of various researchers. Motivated by this structure, we generalize and define a new class of graphs named 'partial chain graphs' and study the properties. We also give the expression for rank, determinant and permanent of these graphs, from which permanent and determinants of well-known wheel graphs, fan graphs, and friendship graphs can be derived.