The Jahn-Teller (JT) theorem predicts spontaneous symmetry breaking and lifting of degeneracy in degenerate electronic states of (nonlinear) molecular and solid-state systems. In these cases, degeneracy is lifted by geometric distortion. Molecular problems are often modelled using spectral theory for weighted graphs, and the present paper turns this process around and reformulates the JT theorem for general vertex- and edge-weighted graphs themselves. If the eigenvectors and eigenvalues of a general graph are considered as orbitals and energy levels (respectively) to be occupied by electrons, then degeneracy of states can be resolved by a non-totally symmetric re-weighting of edges and, where necessary, vertices. This leads to the conjecture that whenever the spectrum of a graph contains a set of bonding or anti-bonding degenerate eigenvalues, the roots of the Hamiltonian matrix over this set will show a linear dependence on edge distortions, which has the effect of lifting the degeneracy. When the degenerate level is non-bonding, distortions of vertex weights have to be included to obtain a full resolution of the eigenspace of the degeneracy. Explicit treatments are given for examples of the octahedral graph, where the degeneracy to be lifted is forced by symmetry, and the phenalenyl graph, where the degeneracy is accidental in terms of the automorphism group.

COBISS.SI-ID: 16090457

In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.

COBISS.SI-ID: 1024390740

Polynomial interpolation can be used to obtain closed formulas for topological indices of infinite series of molecular graphs. The method is discussed and its advantages and limitations are pointed out. This is illustrated on fullerenes ▫$C_{12k+4}$▫ and four topological indices: the Wiener index, the edge Wiener index, the eccentric connectivity index, and the reverse Wiener index. The results for the edge Wiener index correct earlier computations from the literature. Related formulas are derived for cyclic phenylenes.

COBISS.SI-ID: 16309593