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 Cayley graphs of rectangular groups

In this paper, we first give a characterization of Cayley graphs of rectangular groups. Then, vertex-transitivity of Cayley graphs of rectangular groups is considered. Further, it is shown that Cayley graphs Cay(S,C) which are automorphism-vertex-transitive, are in fact Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox semigroup. Finally, a characterization of vertex-transitive graphs which are Cayley graphs of finite semigroups is concluded.     

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 $G$.     

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.     

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.     

On the classification of quartic half-arc-transitive metacirculants

A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1 and n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ^m has at least one fixed vertex. In a recent paper Maruši? and the author showed that each connected quartic half-arc-transitive metacirculant belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph relative to the semiregular automorphism ρ. One of these classes coincides with the class of the so-called tightly-attached graphs, which have already been completely classified. In this paper a complete classification of the second of these classes, that is the class of quartic half-arc-transitive metacirculants for which the quotient graph relative to the semiregular automorphism ρ is a cycle with a loop at each vertex, is given.     

A Generalization of Noncommuting Graph via Automorphisms of a Group

Let G be a group. By using a family 𝒜 of subsets of automorphisms of G, we introduced a simple graph Γ_𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ_𝒜(G).     

Hamilton paths and cycles in vertex-transitive graphs of order

It is shown that every connected vertex-transitive graph of order 6p, where p is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order 6p which is not genuinely imprimitive contains a Hamilton cycle.     

An unusual decomposition of a complete 7-partite graph of order 28

A decomposition of the complete 7-partite graph on 28 vertices where each set in the partition has four vertices is given. Several unusual properties of this decomposition are discussed, giving rise to several natural questions.     

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.     

A note on the automorphism groups of cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph on G is said to be normal if . In this note, we prove that connected cubic non-symmetric Cayley graphs of the ten finite non-abelian simple groups G in the list of non-normal candidates given in [X.G. Fang, C.H. Li, J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math. 244 (2002) 67-75] are normal.