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Register a new userReturn probabilities on nonunimodular transitive graphs
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Pengfei Tang
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Consider simple random walk $(S_n)_{ngeq0}$ on a transitive graph with spectral radius $rho$. Let $u_n=mathbb{P}[S_n=S_0]$ be the $n$-step return probability. It is a folklore conjecture that on transient, transitive graphs $u_n/rho^n$ is at most of the order $n^{-3/2}$. We prove this conjecture for graphs with a closed, transitive, amenable and nonunimodular subgroup of automorphisms. We also study the first return probability $f_n$. For a graph $G$ with a closed, transitive, nonunimodular subgroup of automorphisms, we show that there is a positive constant $c$ such that $f_ngeq frac{u_n}{cn^c}$. We also make some conjectures related to $f_n$ and $u_n$ for transient, transitive graphs.
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Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.
A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers.
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0rightarrow v_1rightarrow cdotsrightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_irightarrow v_j$ is an arc for all $0leq i<jleq k$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via ErdH{o}s theorem by Halld{o}rsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph $K(8,3)$ is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Grotzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvatal graph) are not semi-transitive. Hence, the Grotzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph $C(13;1,5)$ on 13 vertices) and dense ones (Tofts graphs). Finally, we show that each $4$-regular circulant graph (possibly containing triangles) is semi-transitive.
We consider the class of semi-transitively orientable graphs, which is a much larger class of graphs compared to transitively orientable graphs, in other words, comparability graphs. Ever since the concept of a semi-transitive orientation was defined as a crucial ingredient of the characterization of alternation graphs, also knownas word-representable graphs, it has sparked independent interest. In this paper, we investigate graph operations and graph products that preserve semitransitive orientability of graphs. The main theme of this paper is to determine which graph operations satisfy the following statement: if a graph operation is possible on a semitransitively orientable graph, then the same graph operation can be executed on the graph while preserving the semi-transitive orientability. We were able to prove that this statement is true for edge-deletions, edge-additions, and edge-liftings. Moreover, for all three graph operations,we showthat the initial semi-transitive orientation can be extended to the new graph obtained by the graph operation. Also, Kitaev and Lozin explicitly asked if certain graph products preserve the semitransitive orientability. We answer their question in the negative for the tensor product, lexicographic product, and strong product.We also push the investigation further and initiate the study of sufficient conditions that guarantee a certain graph operation to preserve the semi-transitive orientability.
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