On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.     

On the intersection graph of a finite group

For a finite group G, the intersection graph of G which is denoted by Γ(G) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H ∩ K ≠ 1. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut(Γ(G)).     

On the full automorphism group of a graph

While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ _p wrZ _p , almost all Cayley graphs ofG have automorphism group isomorphic toG.     

On the power graph and the reduced power graph of a finite group

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups.     

群与图的对称性

首先对平面图形的对称进行分析和利用其对称群进行量化,进而将此推广到考察一般图的广义"对称性"与图自同构群的关系,最后刻画了无平方因子阶局部本原弧传递图的自同构群结构.     

On the action of a group on a graph, II

This paper is a continuation of the survey by the author (V.I. Trofimov, On the action of a group on a graph, Acta Appl. Math. 29 (1992) 161–170) on some results concerning groups of automorphisms of locally finite vertex-symmetric graphs.     

On the equations of the edge cone of a graph¶and some applications

Let G be a graph such that none of its components is bipartite. We describe the facets of the cone generated by the columns of the incidence matrix of G. Let k[G] be the subring generated by the monomials of degree two defining the edges of G, where k is a field. Some estimates for the a-invariant of k[G] are shown when G is the cone of a normal connected non bipartite graph or G is the join of two normal connected non bipartite graphs. Received: 24 July 1997 / Revised version: 3 March 1998     

On the Sylow graph of a group and Sylow normalizers

Let G be a finite group and G _p be a Sylow p-subgroup of G for a prime p in π(G), the set of all prime divisors of the order of G. The automiser A _p (G) is defined to be the group N _G (G _p )/G _p C _G (G _p ). We define the Sylow graph Γ _A (G) of the group G, with set of vertices π(G), as follows: Two vertices p, q ∈ π(G) form an edge of Γ _A (G) if either q ∈ π(A _p (G)) or p ∈ π(A _q (G)). The following result is obtained Theorem: Let G be a finite almost simple group. Then the graph Γ _A (G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.     

On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k ? 1) + ... + k(k ? 1)^d?1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k ² + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).     

On the generating graph of direct powers of a simple group

Let S be a nonabelian finite simple group and let n be an integer such that the direct product S ⁿ is 2-generated. Let Γ(S ⁿ ) be the generating graph of S ⁿ and let Γ _n (S) be the graph obtained from Γ(S ⁿ ) by removing all isolated vertices. A recent result of Crestani and Lucchini states that Γ _n (S) is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that diam(Γ ₁(S))=2, and we define Δ(S) to be the maximal n such that diam(Γ _n (S))=2. We prove that Δ(S)≥2 for all S, which is best possible since Δ(A ₅)=2, and we show that Δ(S) tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups.     

On the order supergraph of the power graph of a finite group

Ricerche di Matematica - The power graph $${\mathcal {P}}_{G}$$ of a finite group G is the graph whose vertex set is G, two distinct vertices are adjacent if one is a power of the other. The order...     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

On the independence number of the power graph of a finite group

The power graph ${\Gamma }_G$ of a finite group $G$ is the graph whose vertex set is $G$, two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of ${\Gamma }_G$ and characterize the groups achieving the bounds. Moreover, we determine the independence number of ${\Gamma }_G$ if $G$ is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups $G$ whose power graphs have independence number 3 or $n?2$, where $n$ is the order of $G$.