Let Γ = Cay(G, S) and G ≤ X ≤ AutΓ. We say Γ is (X, 1)-regular Cayley graph if X acts regularly on its arcs. Γ is said to be corefree if G is core-free in some X ≤ Aut(Cay(G, S)). In this paper, we prove that if an (X, 1)-regular Cayley graph of valency 5 is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free ...

Let G be a graph and F ⊆ V (G). Then F is called an induced forest of G if 〈G〉 contains no cycle. The forest number, f(G), of G is defined by f(G) := max{|F | : F is an induced forest of G}. It was proved by the second author in [6] that if G is an r-regular graph of order n, then f(G) ≤ b nr−2 2(r−1)c. It was also proved that the bound is sharp by constructing an r-regular graph H of order n w...

in this paper, we investigate various kinds of extensions of twin-good rings. moreover, we prove that every element of an abelian neat ring r is twin-good if and only if r has no factor ring isomorphic to z2  or z3. the main result of [24] states some conditions that any right self-injective ring r is twin-good. we extend this result to any regular baer ring r by proving that every element of a...

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...

We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (logd−1 |G|) /2 and that random d-regular Cayley graphs of simple algebraic groups over Fq asymptotically almost surely have girth at least logd−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|) with α < 1. Several conjectures and open quest...

We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If G is a connected d-regular graph with n vertices, the graph square of G is not complete, and G is not a member of two narrow families of graphs, then the square of G has at least (2− od(1))n more edges than G.

Let Γ(G) be a set of all simple graphs of order n and sizem, without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ·· · ≥ dn > 0. A graph G is regular if and only if d1 = d2 = · · ·= dn. Each mapping Irr : Γ(G) 7→ [0,+∞) with the property Irr(G) = 0 if and only if G is regular, is referred to as irregularity measure of graph. In this paper we introduce some new irregularity measures ...

We investigate the automorphisms of some $$\kappa $$ -existentially closed groups. In particular, we prove that Aut(G) is union subgroups level preserving and $$|Aut(G)|=2^{\kappa }$$ whenever inaccessible G unique group cardinality . Indeed, latter result a byproduct an argument showing that, for any uncountable limit regular representation length with countable base, have $$|Aut(G)|=\beth _{\...

studied by Kazhdan and Lusztig in [KL88]. For x = gG(O) ∈ X the G(O)-orbit (for the adjoint action) of Ad(g)(u) in g(O) depends only on x, and its image under g(O) ։ g(C) is a well-defined G(C)orbit in g(C). We say that x ∈ X is regular if the associated orbit is regular in g(C). (Recall that an element of g(C) is regular if the nilpotent part of its Jordan decomposition is a principal nilpoten...

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...