The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

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 Distance Laplacian Spectra of Certain Finite Groups

Acta Mathematica Sinica, English Series - For a finite group G, the power graph $$\cal{P}(G)$$ is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x...     

关于有限群的强幂图的谱

令$G$是一个阶为$n$的有限群, $G$上的强幂图定义为: 以$G$为顶点集, 对于两个不同的元素$x$和$y$, 如果存在两个不超过$n$的正整数$n_1, n_2$使得$x^{n_1}=y^{n_2}$, 则$x$和$y$ 之间连一条边. 本文给出了$G$上强幂图的距离矩阵和邻接矩阵的特征多项式, 并且计算了其距离谱和邻接谱.     

Isomorphisms of finite semi-Cayley graphs

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.     

The Critical Group of a Line Graph

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph.

• The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of the graph, is shown also to bound the number of generators for the critical group of its line graph.

• The second gives, for each prime p, a constraint on the p-primary structure of the critical group, based on the largest power of p dividing all sums of degrees of two adjacent vertices.

• The third deals with connected graphs whose line graph is regular. Here known results relating the number of spanning trees of the graph and of its line graph are sharpened to exact sequences which relate their critical groups.

The first two results interact extremely well with the third. For example, they imply that in a regular nonbipartite graph, the critical group of the graph and that of its line graph determine each other uniquely in a simple fashion.     

Lambda number of the power graph of a finite group

Journal of Algebraic Combinatorics - The power graph $$\Gamma _G$$ of a finite group G is the graph with the vertex set G, where two distinct elements are adjacent if and only if one is a power of...     

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.     

Finite almost simple groups with prime graphs all of whose connected components are cliques

We find finite almost simple groups with prime graphs all of whose connected components are cliques, i.e., complete graphs. The proof is based on the following fact, which was obtained by the authors and is of independent interest: the prime graph of a finite simple nonabelian group contains two nonadjacent odd vertices that do not divide the order of the outer automorphism group of this group.     

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph’s binary cycle space, each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no non-trivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.