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].

PEWARNAAN TITIK TOTAL SUPER ANTI-AJAIB LOKAL PADA GRAF PETERSEN DIPERUMUM P(n,k) DENGAN k=1,2

The local antimagic total vertex labeling of graph G is a labeling that every vertices and edges label by natural number from 1 to such that every two adjacent vertices has different weights, where is The sum of a vertex label and the labels of all edges that incident to the vertex. If the labeling start the smallest label from the vertex then the edge so that kind of coloring is called the local super antimagic total vertex labeling. That local super antimagic total vertex labeling induces vertex coloring of graph G where for vertex v, the weight w(v) is the color of v. The minimum number of colors that obtained by coloring that induces by local super antimagic total vertex labeling of G called the chromatic number of local super antimagic total vertex coloring of G, denoted by χlsat(G). In this paper, we consider the chromatic number of local super antimagic total vertex coloring of Generalized Petersen Graph P(n,k) for k=1, 2.

Partition Dimension of Generalized Petersen Graph

Let G = V G , E G be the connected graph. For any vertex i ∈ V G and a subset B ⊆ V G , the distance between i and B is d i ; B = min d i , j | j ∈ B . The ordered k -partition of V G is Π = B 1 , B 2 , … , B k . The representation of vertex i with respect to Π is the k -vector, that is, r i | Π = d i , B 1 , d i , B 2 , … , d i , B k . The partition Π is called the resolving (distinguishing) partition if r i | Π ≠ r j | Π , for all distinct i , j ∈ V G . The minimum cardinality of the resolving partition is called the partition dimension, denoted as pd G . In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.

On (2-d)-Kernels in Two Generalizations of the Petersen Graph

A subset J is a (2-d)-kernel of a graph if J is independent and 2-dominating simultaneously. In this paper, we consider two different generalizations of the Petersen graph and we give complete characterizations of these graphs which have (2-d)-kernel. Moreover, we determine the number of (2-d)-kernels of these graphs as well as their lower and upper kernel number. The property that each of the considered generalizations of the Petersen graph has a symmetric structure is useful in finding (2-d)-kernels in these graphs.

The r-Extra Diagnosability of Hyper Petersen Graphs

The diagnosability of a multiprocessor system or an interconnection network plays an important role in measuring the fault tolerance of the network. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the [Formula: see text]-extra diagnosability, which restrains that every fault-free component has at least [Formula: see text] fault-free nodes. As a famous topology structure of interconnection networks, the hyper Petersen graph [Formula: see text] has many good properties. It is difficult to prove the [Formula: see text]-extra diagnosability of an interconnection network. In this paper, we show that the [Formula: see text]-extra diagnosability of [Formula: see text] is [Formula: see text] for [Formula: see text] and [Formula: see text] in the PMC model and for [Formula: see text] and [Formula: see text] in the MM[Formula: see text] model.