Eigenspectral Analysis of Pendant Vertex- and Pendant Edge-Weighted Graphs of Linear Chains, Cycles, and Stars

Abstract Three classes of pendent vertex- and pendant edge-weighted graphs of linear chains (class I), stars (class II), and cycles (class III) have been presented. These graphs (particularly class I and III) represent heteroconjugated π-systems. Sometimes such graphs appear as factored subgraphs of some complicated graphs. The eigenspectra of these graphs have been found out in analytical forms. These eigenspectra have been utilized i) to calculate the band (i.e., HOMO–LUMO) gaps of such graphs, ii) to find out three classes of inversely proportional graphs (with inversely proportional pairs of eigenvalues), and iii) to express eigenspectra of some complicated graphs in analytical forms along with some subspectral relationships. In the limit of n (number of vertices) to infinity, the band gap of the graph of class I has been shown to be the same with that calculated by considering its hypothetical “cyclic dimer.” Reciprocal graphs have also been considered in this context. These graphs are not all hypothetical. A few of them have also been synthesized.