The prime graph of a finite group $G$ is denoted by$ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by primegraph, if for every finite group $H$, where $ga(H)=ga(G)$, thereexists a nonabelian composition factor of $H$ which is isomorphic to$G$. Until now, it is proved that some finite linear simple groups arequasirecognizable by prime graph, for instance, the linear groups $L_...

Families of expander graphs are sparse graphs such that the number of vertices in each graph grows yet each graph remains difficult to disconnect. Expander graphs are of great importance in theoretical computer science. In this paper we study the connection between the Cheeger constant, a measure of the connectivity of the graph, and the smallest nonzero eigenvalue of the graph Laplacian. We sh...

Given a group or semi-group Γ generated (in the sense of semi-groups) by some subset S ⊂ Γ, the Cayley graph of Γ with respect to S is the oriented graph having vertices γ ∈ Γ and oriented edges (γ, γs)s∈S . In this paper, we consider generating sets S which are not necessarily finite thus yielding oriented Cayley graphs which are perhaps not locally finite. The above situation gives rise to a ...

We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decompositi...

Let ∆ be a finite building (or, more generally, a thick spherical and locally finite building). The chamber graph G(∆), whose edges are the pairs of adjacent chambers in ∆, is known to be q-regular for a certain number q = q(∆). Our main result is that G(∆) is q-connected in the sense of graph theory. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes...

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodesic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodesic cycles need not generate C(G).

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G).

let $g$ be a finite group. we construct the prime graph of $ g $,which is denoted by $ gamma(g) $ as follows: the vertex set of thisgraph is the prime divisors of $ |g| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ g $ contains anelement of order $ pq $.in this paper, we determine finite groups $ g $ with $ gamma(g) =gamma(l_3(q)) $, $2 leq q < 100 $ and prov...

‎we pursue further our‎ ‎investigation‎, ‎begun in [h.~smith‎, ‎groups with all subgroups subnormal or‎ ‎nilpotent-by-{c}hernikov‎, ‎emph{rend‎. ‎sem‎. ‎mat‎. ‎univ‎. ‎padova}‎ ‎126 (2011)‎, ‎245--253] and continued in [g.~cutolo and h.~smith‎, ‎locally finite groups with all subgroups‎ ‎subnormal or nilpotent-by-{c}hernikov‎. ‎emph{centr‎. ‎eur‎. ‎j‎. ‎math.} (to appear)] ‎‎‎of groups $g$ in w...