Related papers: Note on uniformly transient graphs
For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class.…
A non-complete graph is \emph{$2$-distance-transitive} if, for $i=1,2$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance $i$ in the graph, there exists an element of the graph automorphism group that maps…
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number.…
We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the…
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann…
The edge betweenness centrality of an edge is loosely defined as the fraction of shortest paths between all pairs of vertices passing through that edge. In this paper, we investigate graphs where the edge betweenness centrality of edges is…
Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for…
We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary conditions on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Polya, Yau, and others.…
A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. This paper shows how the spectrum of $T_n$ can be obtained using the spectral properties of the Jucys-Murphy…
Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group S_n generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The…
We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we…
A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient…
We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated…
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…
Random graph models have played a dominant role in the theoretical study of networked systems. The Poisson random graph of Erdos and Renyi, in particular, as well as the so-called configuration model, have served as the starting point for…
In recent work, we study certain Cayley graphs associated with a finite commutative ring and their multiplicative subgroups. Among various results that we prove, we provide the necessary and sufficient conditions for such a Cayley graph to…
A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected…
In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the…
A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…
The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a ``transmission`` system. A transmission system is a mathematical representation of a means of transmitting…