scholarly journals On Cyclic-Vertex Connectivity of n , k -Star Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Yalan Li ◽  
Shumin Zhang ◽  
Chengfu Ye
Keyword(s):  
Connected Graph ◽  
Vertex Connectivity ◽  
Star Network ◽  
Star Graphs

A vertex subset F   ⊆   V G is a cyclic vertex-cut of a connected graph G if G − F is disconnected and at least two of its components contain cycles. The cyclic vertex-connectivity κ c G is denoted as the cardinality of a minimum cyclic vertex-cut. In this paper, we show that the cyclic vertex-connectivity of the n , k -star network S n , k is κ c S n , k = n + 2 k − 5 for any integer n ≥ 4 and k ≥ 2 .

A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs

2019 ◽  
Vol 63 (9) ◽  
pp. 1372-1384
Author(s):  
Zuwen Luo ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Connected Graph ◽  
Cayley Graphs ◽  
Network Structures ◽  
Vertex Connectivity ◽  
Wheel Graphs

Abstract Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$-vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$. The cardinality of a minimum $R^{k}$-vertex-cut is the $R^{k}$-vertex-connectivity of $G$ and is denoted by $\kappa ^{k}(G)$. $R^{k}$-vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$-vertex-connectivity and $R^{2}$-vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $\kappa ^{1}(AW_{n})=4n-7$ for $n\geq 6$; $\kappa ^{2}(AW_{n})=6n-12$ for $n\geq 6$.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

General eccentric distance sum of graphs

2020 ◽  
pp. 2150046
Author(s):  
Tomáš Vetrík
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Vertex Set ◽  
General Graphs ◽  
Eccentric Distance

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

Connectivity

2013 ◽  
pp. 37-47
Author(s):  
Mahtab Hosseininia ◽  
Faraz Dadgostari
Keyword(s):  
Connected Graph ◽  
Directed Graphs ◽  
Graph Connectivity ◽  
Breadth First Search ◽  
Undirected Graphs ◽  
Vertex Connectivity ◽  
Edge Connectivity ◽  
Graph Traversal ◽  
Path Component ◽  
Block Tree

In this chapter, the concept of graph connectivity is introduced. In the first section, some concepts such as walk, path, component and connected graph are defined, and connectedness of a graph from the viewpoint of vertex connectivity, and also, edge connectivity are discussed. Then, blocks and block tree of graphs are illustrated. In addition, connectivity in directed graphs is introduced. Furthermore, in the last section, two graph traversal algorithms, depth first search and breadth first search, are described to investigate the connectedness of directed and undirected graphs.

Vertex-connectivity in periodic graphs and underlying nets of crystal structures

2016 ◽  
Vol 72 (3) ◽  
pp. 376-384 ◽  
Author(s):  
Jean-Guillaume Eon
Keyword(s):  
Crystal Structures ◽  
Long Range ◽  
Connected Graph ◽  
Three Dimensional ◽  
Topological Properties ◽  
Combinatorial Topology ◽  
Vertex Connectivity ◽  
Quotient Graph ◽  
Periodic Graphs ◽  
Definition Of

Periodic nets used to describe the combinatorial topology of crystal structures have been required to be 3-connected by some authors. A graph isn-connected when deletion of less thannvertices does not disconnect it.n-Connected graphs area fortiarin-coordinated but the converse is not true. This article presents an analysis of vertex-connectivity in periodic graphs characterized through their labelled quotient graph (LQG) and applied to a definition of underlying nets of crystal structures. It is shown that LQGs ofp-periodic graphs (p≥ 2) that are 1-connected or 2-connected, but not 3-connected, arecontractiblein the sense that they display, respectively, singletons or pairs of vertices separating dangling or linker components with zero net voltage over every cycle. The contraction operation that substitutes vertices and edges, respectively, for dangling components and linkers yields a 3-connected graph with the same periodicity. 1-Periodic graphs can be analysed in the same way through their LQGs but the result may not be 3-connected. It is claimed that long-range topological properties of periodic graphs are respected by contraction so that contracted graphs can represent topological classes of crystal structures, be they rods, layers or three-dimensional frameworks.

scholarly journals On the Connectivity of Graphs in Association Schemes

10.37236/6820 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Brian G. Kodalen ◽  
William J. Martin
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Association Scheme ◽  
Undirected Graph ◽  
Association Schemes ◽  
Vertex Connectivity ◽  
Edge Connectivity

Let $(X,\mathcal{R})$ be a commutative association scheme and let $\Gamma=(X,R\cup R^\top)$ be a connected undirected  graph where $R\in \mathcal{R}$. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of $\Gamma$ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two vertices $a,b\in X$  are called "twins" in $\Gamma$ if they have identical neighborhoods: $\Gamma(a)=\Gamma(b)$. We characterize twins in polynomial association schemes and show that, in the absence of twins, the deletion of any vertex and its neighbors in $\Gamma$ results in a connected graph. Using this and other tools, we find lower bounds on the connectivity of $\Gamma$, especially in the case where $\Gamma$ has diameter two.

ON FINDING SPARSE THREE-EDGE-CONNECTED AND THREE-VERTEX-CONNECTED SPANNING SUBGRAPHS

2014 ◽  
Vol 25 (03) ◽  
pp. 355-368
Author(s):  
AMR ELMASRY ◽  
YUNG H. TSIN
Keyword(s):  
Connected Graph ◽  
Sparse Graph ◽  
Input Graph ◽  
Vertex Connectivity ◽  
Time Efficiency ◽  
Depth First Search ◽  
Spanning Subgraphs ◽  
Spanning Subgraph ◽  
Ear Decomposition ◽  
The Given

We present algorithms that construct a sparse spanning subgraph of a three-edge-connected graph that preserves three-edge connectivity or of a three-vertex-connected graph that preserves three-vertex connectivity. Our algorithms are conceptually simple and run in O(|E|) time. These simple algorithms can be used to improve the efficiency of the best-known algorithms for three-edge and three-vertex connectivity and their related problems, by preprocessing the input graph so as to trim it down to a sparse graph. Afterwards, the original algorithms run in O(|V|) instead of O(|E|) time. Our algorithms generate an adjacency-lists structure to represent the sparse spanning subgraph, so that when a depth-first search is performed over the subgraph based on this adjacency-lists structure it actually traverses the paths in an ear-decomposition of the subgraph. This is useful because many of the existing algorithms for three-edge- or three-vertex connectivity and their related problems are based on an ear-decomposition of the given graph. Using such an adjacency-lists structure to represent the input graph would greatly improve the run-time efficiency of these algorithms.

Simpler Sequential and Parallel Biconnectivity Augmentation in Trees

2015 ◽  
Vol 25 (04) ◽  
pp. 1550010 ◽  
Author(s):  
Surabhi Jain ◽  
N. Sadagopan
Keyword(s):  
Parallel Algorithm ◽  
Connected Graph ◽  
Sequential Algorithm ◽  
Vertex Connectivity ◽  
Processor Time ◽  
Vertex Separator ◽  
Edge Contraction ◽  
Pram Model ◽  
Connectivity Augmentation ◽  
The One

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components and a minimum vertex separator is a vertex separator of least cardinality. The vertex connectivity refers to the size of a minimum vertex separator. For a connected graph G with vertex connectivity [Formula: see text], the connectivity augmentation refers to a set S of edges whose augmentation to G increases its vertex connectivity by one. A minimum connectivity augmentation of G is the one in which S is minimum. In this paper, we focus our attention on biconnectivity augmentation for trees. Towards this end, we present a new sequential algorithm for biconnectivity augmentation in trees by simplifying the algorithm reported in [1]. The simplicity is achieved with the help of edge contraction tool. This tool helps us in getting a recursive subproblem preserving all connectivity information. Subsequently, we present a parallel algorithm to obtain a minimum biconnectivity augmentation set in trees. Our parallel algorithm essentially follows the overall structure of sequential algorithm. Our implementation is based on CREW PRAM model with [Formula: see text] processors, where [Formula: see text] refers to the maximum degree of a tree. We also show that our parallel algorithm is optimal with a processor-time product of [Formula: see text] where n is the number of vertices of a tree.

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