Connectivity, genus, and the number of components in vertex-deleted subgraphs

AbstractLet c(G) denote the number of components in a graph G. It is shown that if G has genus γ and isk-connected with k ⩾ 3, then c(G − X) ⩽ (2/(k − 2))(| X | − 2 + 2γ), for all X⊆ V(G) with | X | ⩾ k. Some implications of this result for planar graphs (y = 0) and toroidal graphs (y = 1) are considered

Topological Graph Polynomials in Colored Group Field Theory

In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [arXiv:0907.2582]. We define the boundary graph \cG_{\partial} of an open graph \cG and prove it is a cellular complex. Using this structure we generalize the topological (Bollobas-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension

Modularity of minor-free graphs

We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.Comment: 7 pages, 1 figur