The Generalized Connectivity of Generalized Petersen Graph

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

The Bounds of Generalized 4-Connectivity of Alternating Group Graphs

Generalized connectivity is a parameter of evaluating the reliability of a network. Let [Formula: see text] be a vertex set of graph [Formula: see text] and [Formula: see text], there is a tree in [Formula: see text] as it connects all vertices of [Formula: see text] which is called a [Formula: see text]-tree. Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-trees in [Formula: see text]. If [Formula: see text] and [Formula: see text] are given, we define that two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint. Then the maximum number of internally disjoint [Formula: see text]-trees in [Formula: see text] is denoted by [Formula: see text]. For an integer [Formula: see text] with [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is denoted as [Formula: see text]. In this paper, we study the generalized [Formula: see text]-connectivity of alternating group graphs, denoted by [Formula: see text]. We show the upper bound and the lower bound of [Formula: see text], [Formula: see text] for [Formula: see text], and that [Formula: see text] is given.

The Generalized Connectivity of Bubble-Sort Star Graphs

The connectivity plays an important role in measuring the fault tolerance and reliability of interconnection networks. The generalized [Formula: see text]-connectivity of a graph [Formula: see text], denoted by [Formula: see text], is an important indicator of a network’s ability for fault tolerance and reliability. The bubble-sort star graph, denoted by [Formula: see text], is a well known interconnection network. In this paper, we show that [Formula: see text] for [Formula: see text], that is, for any three vertices in [Formula: see text], there exist [Formula: see text] internally disjoint trees connecting them in [Formula: see text] for [Formula: see text], which attains the upper bound of [Formula: see text] given by Li et al. for [Formula: see text].

The Generalized Connectivity of Data Center Networks

The generalized [Formula: see text]-connectivity of a graph [Formula: see text] is a parameter that can measure the reliability of a network [Formula: see text] to connect any [Formula: see text] vertices in [Formula: see text], which is a generalization of traditional connectivity. Let [Formula: see text] and [Formula: see text] denote the maximum number [Formula: see text] of edge-disjoint trees [Formula: see text] in [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text]. For an integer [Formula: see text] with [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. Data centers are essential to the business of companies such as Google, Amazon, Facebook and Microsoft et al. Based on data centers, the data center networks [Formula: see text], introduced by Guo et al. in 2008, have many desirable properties. In this paper, we study the generalized [Formula: see text]-connectivity of [Formula: see text] and show that [Formula: see text] for [Formula: see text] and [Formula: see text].