scholarly journals Graph connectivity and Wiener index

2006 ◽  
Vol 133 (31) ◽  
pp. 1-5 ◽  
Author(s):  
I. Gutman ◽  
S. Zhang
Keyword(s):  
Complete Graph ◽  
Wiener Index ◽  
Graph Connectivity ◽  
Subject Classification ◽  
Edge Connectivity

The graphs with a given number n of vertices and given (vertex or edge) connectivity k, having minimum Wiener index are determined. In both cases this is Kk + (K1 U Kn-k-1), the graph obtained by connecting all vertices of the complete graph Kk with all vertices of the graph whose two components are Kn-k-1 and K1. AMS Mathematics Subject Classification (2000): 05C12, 05C40 05C35.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Symmetry Breaking in Graphs

10.37236/1242 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Karen L. Collins
Keyword(s):  
Symmetry Breaking ◽  
Complete Graph ◽  
Subject Classification ◽  
Distinguishing Number

A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

ON THE WIENER INDEX OF F_H SUMS OF GRAPHS

2021 ◽  
Vol 3 (2) ◽  
pp. 37-57
Author(s):  
L. Alex ◽  
Indulal G
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Chemical Properties ◽  
Wiener Index ◽  
Topological Indices ◽  
Single Vertex ◽  
Molecular Graphs ◽  
New Class ◽  
And Chemical Properties

Wiener index is the first among the long list of topological indices which was used to correlate structural and chemical properties of molecular graphs. In \cite{Eli} M. Eliasi, B. Taeri defined four new sums of graphs based on the subdivision of edges with regard to the cartesian product and computed their Wiener index. In this paper, we define a new class of sums called $F_H$ sums and compute the Wiener index of the resulting graph in terms of the Wiener indices of the component graphs so that the results in \cite{Eli} becomes a particular case of the Wiener index of $F_H$ sums for $H = K_1$, the complete graph on a single vertex.

scholarly journals Generalized Fuzzy Graph Connectivity Parameters with Application to Human Trafficking

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 424
Author(s):  
Arya Sebastian ◽  
John N Mordeson ◽  
Sunil Mathew
Keyword(s):  
Human Trafficking ◽  
Network Theory ◽  
Graph Connectivity ◽  
Fuzzy Graph ◽  
Graph Models ◽  
Edge Connectivity ◽  
Clustering Techniques ◽  
New Class ◽  
Large Size ◽  
Fuzzy Graphs

Graph models are fundamental in network theory. But normalization of weights are necessary to deal with large size networks like internet. Most of the research works available in the literature have been restricted to an algorithmic perspective alone. Not much have been studied theoretically on connectivity of normalized networks. Fuzzy graph theory answers to most of the problems in this area. Although the concept of connectivity in fuzzy graphs has been widely studied, one cannot find proper generalizations of connectivity parameters of unweighted graphs. Generalizations for some of the existing vertex and edge connectivity parameters in graphs are attempted in this article. New parameters are compared with the old ones and generalized values are calculated for some of the major classes like cycles and trees in fuzzy graphs. The existence of super fuzzy graphs with higher connectivity values are established for both old and new parameters. The new edge connectivity values for some wider classes of fuzzy graphs are also obtained. The generalizations bring substantial improvements in fuzzy graph clustering techniques and allow a smooth theoretical alignment. Apart from these, a new class of fuzzy graphs called generalized t-connected fuzzy graphs are studied. An algorithm for clustering the vertices of a fuzzy graph and an application related to human trafficking are also proposed.

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.

The Restricted Edge-Connectivity of Kronecker Product Graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 1950012
Author(s):  
Tianlong Ma ◽  
Jinling Wang ◽  
Mingzu Zhang
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Kronecker Product ◽  
Sufficient Condition ◽  
Connected Component ◽  
Edge Connectivity ◽  
Product Graphs ◽  
Minimum Number ◽  
Vertex Set ◽  
Product Of Graphs

The restricted edge-connectivity of a connected graph [Formula: see text], denoted by [Formula: see text], if exists, is the minimum number of edges whose deletion disconnects the graph such that each connected component has at least two vertices. The Kronecker product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, it is proved that [Formula: see text] for any graph [Formula: see text] and a complete graph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is minimum edge-degree of [Formula: see text], and a sufficient condition such that [Formula: see text] is [Formula: see text]-optimal is acquired.

scholarly journals Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

2009 ◽  
Vol 22 (10) ◽  
pp. 1571-1576 ◽  
Author(s):  
A. Behtoei ◽  
M. Jannesari ◽  
B. Taeri
Keyword(s):  
Wiener Index ◽  
Graph Connectivity ◽  
Zagreb Index

scholarly journals Extremal topological indices for graphs of given connectivity

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1639-1643 ◽  
Author(s):  
Ioan Tomescu ◽  
Misbah Arshad ◽  
Muhammad Jamil
Keyword(s):  
Wiener Index ◽  
Topological Indices ◽  
Connectivity Index ◽  
Zeroth Order ◽  
Edge Connectivity ◽  
Connected Graphs ◽  
Randic Index ◽  
Randić Connectivity Index ◽  
Unique Graph ◽  
General Randić Connectivity Index

In this paper, we show that in the class of graphs of order n and given (vertex or edge) connectivity equal to k (or at most equal to k), 1 ? k ? n - 1, the graph Kk + (K1? Kn-k-1) is the unique graph such that zeroth-order general Randic index, general sum-connectivity index and general Randic connectivity index are maximum and general hyper-Wiener index is minimum provided ? > 1. Also, for 2-connected (or 2-edge connected) graphs and ? > 0 the unique graph minimizing these indices is the n-vertex cycle Cn.

COMMON MULTIPLES OF COMPLETE GRAPHS

2003 ◽  
Vol 86 (2) ◽  
pp. 302-326 ◽  
Author(s):  
DARRYN BRYANT ◽  
BARBARA MAENHAUT
Keyword(s):  
Complete Graph ◽  
Subject Classification ◽  
Complete Graphs ◽  
Steiner Triple Systems ◽  
Common Multiple ◽  
Triple Systems ◽  
Johnson Graphs ◽  
Positive Integers

A graph $H$ is said to divide a graph $G$ if there exists a set $S$ of subgraphs of $G$, all isomorphic to $H$, such that the edge set of $G$ is partitioned by the edge sets of the subgraphs in $S$. Thus, a graph $G$ is a common multiple of two graphs if each of the two graphs divides $G$. This paper considers common multiples of a complete graph of order $m$ and a complete graph of order $n$. The complete graph of order $n$ is denoted $K_n$. In particular, for all positive integers $n$, the set of integers $q$ for which there exists a common multiple of $K_3$ and $K_n$ having precisely $q$ edges is determined.It is shown that there exists a common multiple of $K_3$ and $K_n$ having $q$ edges if and only if $q \equiv 0 \, ({\rm mod}\, 3)$, $q \equiv 0 \, ({\rm mod}\, \binom n2)$ and(1) $q \neq 3 \binom n2$ when $n \equiv 5 \, ({\rm mod}\, 6)$; (2) $q \geq (n + 1) \binom n2$ when $n$ is even; (3) $q \notin \{36, 42, 48\}$ when $n = 4$.The proof of this result uses a variety of techniques including the use of Johnson graphs, Skolem and Langford sequences, and equitable partial Steiner triple systems.2000 Mathematical Subject Classification: 05C70, 05B30, 05B07.

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