On several symmetry conditions for graphs

Distance-balanced graphs: Symmetry conditions

A graph X is said to be distance–balanced if for any edge uv of X , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance–balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k + 1 from u a...

Extension of several sufficient conditions for Hamiltonian graphs

Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X\{u})| + d(u) ≥ n − 1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C7 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph (Kr ∪ Ks ∪ Kt) ∨ K2, 1 ≤ r ...

A Symmetry Index for Graphs

This note introduces a quantitative measure (S(G)) of the symmetry structure of a graph G. S(G) is called the index of symmetry of G and is a function of its structural information content and the order of its automorphism group. Properties of S(G) are examined and applications to special classes of graphs are presented. In particular, the index of symmetry of a class of alkanes is shown to be ...

Bohr-Sommerfeld conditions for several commuting Hamiltonians

On minimal imperfect graphs with circular symmetry

Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge’s famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (”odd holes”) and their complements (”odd antiholes”). Only two constructions (due to Chvátal, Graham, Pero...