Counterexamples to a conjecture of Dias on eigenvalues of chemical graphs (Q2853191)

Chemical graphs are simple, connected graphs with maximum degree not greater than three. The limitation on the vertex degree confines the eigenvalues \(\{\lambda_i\}\) of a graph of order \(n\) to the range \(-3\leq\lambda_i\leq3\). The integer nature of the coefficients of the characteristic polynomials implies that, for integers \(p\) that are not squares, the spectrum of a given chemical graph includes either both \(+\sqrt{p}\) and \(-\sqrt{p}\) or neither.NEWLINENEWLINE Chemical graphs with adjacency eigenvalues \(\pm\sqrt{1}\), \(\pm\sqrt{2}\), \(\pm\sqrt{3}\), \(\pm\sqrt{4}\), \(\pm\sqrt{5}\), \(\pm\sqrt{6}\), \(\pm\sqrt{7}\), \(\pm\sqrt{9}\) are all found. The present note proves that at least one chemical graph with eigenvalues \(\pm\sqrt{8}\), called octo-graph, exists for all orders \(n\geq10\). In fact, the eigenvalue pair can be achieved with arbitrarily high multiplicity.