‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

the non-commuting graph $nabla(g)$ of a non-abelian group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we 'll prove that if $g$ is a finite group with $nabla(g)congnabla(bs_{n})$, then $g cong bs_{n}$, where $bs_{n}$ is the symmetric group of degre...

We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include “Can the Baire measurable chromatic number of a locally finite exceed usual by more than one?” and marked groups with isomorphic Cayley have numbers for their shift differ also provide new bound whose connected components all two ends.

A counterexample is presented to the following conjecture of Jackson and Wormald: If j ~ 1, k ~ 2 and a graph is connected, locally jconnected and K l , (j+l)(k-l)+2 -free then it has a k-tree. Preliminaries All graphs considered here are finite and without loops or multiple edges. As usual, we let V(G) and E(G) denote respectively the vertex set and the edge set of the graph G. The cardinality...