Cluster capacity functionals and isomorphism theorems for Gaussian free fields

We investigate level sets of the Gaussian free field on continuous transient metric graphs $$\widetilde{{\mathcal {G}}}$$ G ~ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph $${\mathcal {G}}$$ G , that the capacity of sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on $$\widetilde{{\mathcal {G}}}$$ G ~ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman’s refinement of Lupu’s recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on $$\widetilde{{\mathcal {G}}}$$ G ~ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.

Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

Presentations for vertex-transitive graphs

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).