Let X be a connected, locally finite graph with symmetric growth. We prove that there is vertex coloring ϕ : → {0, 1} and some R ∈ ℝ such every automorphism f preserving -close to the identity map; this can seen as coarse geometric version of symmetry breaking. also infinite motion conjecture true for graphs where at least one stabilizer S x satisfies following condition: non-identity , sequenc...

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify connected heptavalent symmetric graphs of order 6p for each prime p. As a result, there are three sporadic such graphs: one for p = 5 and two for p = 13.

In [2] (and earlier in [3]), we defined several global communication tasks (universal exchange, universal broadcast, universal summation) on directed or vertex symmetric networks. In that paper, we mainly focused on universal broadcast. In [4], we discussed universal sum. Here we continue this work and focus on universal exchange (transpose) in directed vertex symmetric graphs. Introduction. In...

Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex trans...

In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite sequenceM1,M2, . . . of (skew) symmetric matrices over F of bounded F-rank-width has a pair i < j, such that Mi is isomorphic to a principal submatrix of a principal pivot transform of Mj . We generalise this result to σ-symmetric matrices ...

Condit ions are given for the existence of hamiltonian paths and cycles in the so-called Toeplitz graphs, i.e. simple graphs with a symmetric Toeplitz adjacency matrix.