Characteristic Polynomial Followed by Trigonometric Identity for Obtaining Analytical Eigenspectra of Some Weighted Graphs of Linear Chains and Cycles

Abstract

Abstract Graphs of linear chains and/or cycles with homogeneous and/or alternant vertex weights have been considered. If one puts weight(s) on the terminal vertex (vertices) or edge(s) of such linear chains, another set of linear chains might result. All such vertex- or edge-weighted graphs of linear chains and cycles represent conjugated systems with heteroatom(s). Characteristic polynomial followed by trigonometric identity has been shown to be used for obtaining eigenspectra for such systems of linear chains and cycles in closed analytical forms. The analytical expressions so obtained for such systems can be used for deriving the eigenspectra of some complicated π-conjugated molecules in analytical forms.