Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains, cycles and stars

Abstract

Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such pendant graphs become reciprocal graphs for linear chains and cycles if p = n. The n-vertex star graphs follow the same rule as paths and cycles, they become reciprocal on adding a pendant vertex to each of n vertices. The formulas so developed have been expressed in matrix product and in analytical forms for the three classes of graphs that require only the values of n and p for calculation of the respective CP coefficients. Such formulas have the general applicability for a large variety of molecular graphs with varying n and p and have been shown to be reduced to the corresponding formulas for reciprocal graphs that are the special cases of the graphs discussed here.

Keywords:

• characteristic polynomial
• pendant graphs
• reciprocal graphs

Acknowledgements

Authors are for thankful to the UGC, New Delhi, for providing financial assistance extended through the CAS, Department of Chemistry, The University of Burdwan. Authors are also thankful to the learned reviewers for their helpful comments.