Lambda number of the power graph of a finite group
The power graph of a finite group is the graph with the vertex set , where two distinct elements are adjacent if one is a power of the other. An -labeling of a graph is an assignment of labels from nonnegative integers to all vertices of such that vertices at distance two get different labels and adjacent vertices get labels that are at least apart. The lambda number of , denoted by , is the minimum span over all -labelings of . In this paper, we obtain bounds for , and give necessary and sufficient conditions when the bounds are attained. As applications, we compute the exact value of if is a dihedral group, a generalized quaternion group, a -group or a cyclic group of order , where and are distinct primes and is a positive integer.
Keywords: Power graph, -labeling, -number, finite group
MSC 2010: 05C25, 05C78
All graphs considered in this paper are finite, simple and undirected. Let be a graph with the vertex set and the edge set . The distance between vertices and is the length of a shortest path from to in . For nonnegative integers and , an -labeling of is a nonnegative integer valued function on such that whenever and are vertices of distance two and whenever and are adjacent. The span of is the difference between the maximum and minimum values of . The -labeling number of is the minimum span over all -labelings of . The classical work of the -labeling problem is when and . The -labeling number of a graph is also called the -number of and denoted by .
The problem of studying -labelings of a graph is motivated by the radio channel assignment problem [Ha] and by the study of the scalability of optical networks [Rob]. In 1992, Griggs and Yeh [GY92] formally introduced the notion of the -labeling of a graph, and showed that the -labeling problem is NP-complete for general graphs. The -labelling problem, in particular in the case, has been studied extensively; see [Ge, Ge2, W1, KiM] for examples. Surveys of results and open questions on the -labeling problem can be found in [Ye].
Graphs associated with groups and other algebraic structures have been actively investigated, since they have valuable applications (cf. [KeR]) and are related to automata theory (cf. [K, K1]). Zhou [Zh1] studied -labelings of Cayley graphs of abelian groups. Kelarev, Ras and Zhou [Kerz] established connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. In this paper we study -labelings of the power graph of a finite group.
The undirected power graph of a finite group has the vertex set and two distinct elements are adjacent if one is a power of the other. The concepts of a power graph and an undirected power graph were first introduced by Kelarev and Quinn [n1] and by Chakrabarty et al. [CGS], respectively. Since this paper deals only with undirected graphs, we use the term “power graph” to refer to an undirected power graph. Many interesting results on power graphs have been obtained in [Cam, CGh, FMW, FMW1, kel2, MF, MFW, man]. A detailed list of results and open questions on power graphs can be found in [AKC].
Section 2 gives some preliminary results. In Section 3, we obtain a sharp lower bound for the -number of the power graph of a finite group ; as applications, we compute if is a dihedral group, a generalized quaternion group or a -group. In Section 4, we construct an upper bound for , and classify all groups such that the upper bound is attained.
A path covering of a graph , denoted by , is a collection of vertex-disjoint paths in such that each vertex in is contained in a path in . The path covering number of is the minimum cardinality of a path covering of . Let denote the complement of .
([Ge, Theorem 1.1]) Let be a graph of order .
(i) Then if and only if .
(ii) Let be an integer at least . Then if and only if .
A vertex is a cut vertex in a grph if contains more connected components than does, where is the graph obtained by deleting the vertex from .
Let be a graph of order with a cut vertex . Suppose that all connected components of are and for , where . If then .
Note that . If , then is a path, and so . In the following, suppose . Write , and pick . Assume for some . Then
where is the degree of in . Since , we have , which implies that . It follows from Dirac’s theorem ([Bondy, Theorem 4.3]) that has a Hamilton cycle, and so . Note that . Then . By Proposition 2.1, we get the desired result. ∎
Let be a graph. A subset of is a clique if any two distinct vertices in this subset are adjacent in . The clique number is the maximum cardinality of a clique in . It is easy to see that
We give a sufficient condition for reaching the lower bound in (1).
Let be a clique of a graph such that . Then if there exist partitions
of and , respectively, satisfying the follows for each index .
(ii) Every vertex in and every vertex in are nonadjacent in .
- Corresponding author.
- E-mail addresses: email@example.com (X. Ma), firstname.lastname@example.org (M. Feng),
email@example.com (K. Wang).