Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT

Three classes of reciprocal graphs, viz. monocycle (G^C_n), linear chain (G^L_n) and star (G^K_n) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (G^C_n) and linear chain (G^L_n) are obtained by putting a pendant vertex to each vertex of simple monocycle (C_n) and simple linear chain (L_n), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n ? 1) peripheral vertices of the star graph (K_{1, (n?1)}). An n-fold rotational axis of symmetry for G^C_n and (n ? 1)-fold rotational axis of symmetry for G^K_n have been exploited for obtaining their respective condensed graphs. The condensed graph for G^L_n has been generated from that of G^C_n incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.