scholarly journals Connectedness criteria for graphs by means of omega invariant

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 647-652
Author(s):  
Utkum Sanli ◽  
Feriha Celik ◽  
Sadik Delen ◽  
Ismail Cangul
Keyword(s):  
Graph Theory ◽  
Euler Characteristic ◽  
Connected Graph ◽  
Topological Property ◽  
Degree Sequence ◽  
Direct Information ◽  
Combinatorial Properties ◽  
Cyclomatic Number ◽  
The Common ◽  
Disconnected Graph

A realizable degree sequence can be realized in many ways as a graph. There are several tests for determining realizability of a degree sequence. Up to now, not much was known about the common properties of these realizations. Euler characteristic is a well-known characteristic of graphs and their underlying surfaces. It is used to determine several combinatorial properties of a surface and of all graphs embedded onto it. Recently, last two authors defined a number ? which is invariant for all realizations of a given degree sequence. ? is shown to be related to Euler characteristic and cyclomatic number. Several properties of ? are obtained and some applications in extremal graph theory are done by authors. As already shown, the number gives direct information compared with the Euler characteristic on the realizability, number of realizations, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc. In this paper, another important topological property of graphs which is connectedness is studied by means of ?. It is shown that all graphs with ?(G)?-4 are disconnected, and if ?(G)? -2, then the graph could be connected or disconnected. It is also shown that if the realization is a connected graph and ?(G)=-2, then certainly the graph should be acyclic. Similarly, it is shown that if the realization is a connected graph G and ?(G)? 0, then certainly the graph should be cyclic. Also, the fact that when ?(G)?-4, the components of the disconnected graph could not all be cyclic, and that if all the components of a graph G are cyclic, then ?(G) ? 0 are proven.

scholarly journals The effect of edge and vertex deletion on omega invariant

2020 ◽  
pp. 46-46 ◽  
Author(s):  
Sadik Delen ◽  
Muge Togan ◽  
Aysun Yurttas ◽  
Ugur Ana ◽  
Ismail Cangu
Keyword(s):  
Euler Characteristic ◽  
Degree Sequence ◽  
Graph Invariant ◽  
Combinatorial Properties ◽  
Number Of Components ◽  
Vertex Deletion ◽  
The Family

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations. In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals The Partition Dimension of a Path Graph

2021 ◽  
Vol 13 (2) ◽  
pp. 66
Author(s):  
Vivi Ramdhani ◽  
Fathur Rahmi
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Ordered Set ◽  
Minimum Cardinality ◽  
Path Graph ◽  
Partition Dimension

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph,  with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph  and  is a subset of  writen . Suppose there is , then the distance between and  is denoted in the form . There is an ordered set of -partitions of, writen then  the representation of with respect tois the  The set of partitions ofis called a resolving partition if the representation of each  to  is different. The minimum cardinality of the solving-partition to  is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with ,  and  are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

scholarly journals On the Edge Metric Dimension of Certain Polyphenyl Chains

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Muhammad Ahsan ◽  
Zohaib Zahid ◽  
Dalal Alrowaili ◽  
Aiyared Iampan ◽  
Imran Siddique ◽  
...  
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Metric Dimension ◽  
Chain Graph ◽  
Minimum Number ◽  
Valence Bonds

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex w and an edge f = c 1 c 2 of a connected graph G , the minimum number from distances of w with c 1 and c 2 is called the distance between w and f . If for every two distinct edges f 1 , f 2 ∈ E G , there always exists w 1 ∈ W E ⊆ V G such that d f 1 , w 1 ≠ d f 2 , w 1 , then W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph O n , meta-polyphenyl chain graph M n , and the linear [n]-tetracene graph T n and also find the edge metric dimension of para-polyphenyl chain graph L n . It has been proved that the edge metric dimension of O n , M n , and T n is bounded, while L n is unbounded.

scholarly journals Erdos – Renyi Random Graph and Machine Learning Based Botnet Detection

2020 ◽  
Vol 9 (5) ◽  
pp. 1037-1040
Keyword(s):  
Machine Learning ◽  
Probability Theory ◽  
Graph Theory ◽  
Complex Network ◽  
Random Graphs ◽  
Connected Graph ◽  
Free Graph ◽  
Botnet Detection ◽  
Scale Free ◽  
Made In

Basically large networks are prone to attacks by bots and lead to complexity. When the complexity occurs then it is difficult to overcome the vulnerability in the network connections. In such a case, the complex network could be dealt with the help of probability theory and graph theory concepts like Erdos – Renyi random graphs, Scale free graph, highly connected graph sequences and so on. In this paper, Botnet detection using Erdos – Renyi random graphs whose patterns are recognized as the number of connections that the vertices and edges made in the network is proposed. This paper also presents the botnet detection concepts based on machine learning.

scholarly journals The Combinatorics of Hyperbolized Manifolds

2015 ◽  
Vol 117 (1) ◽  
pp. 31
Author(s):  
Allan L. Edmonds ◽  
Steven Klee
Keyword(s):  
Euler Characteristic ◽  
Generating Functions ◽  
Characteristic Sign ◽  
Combinatorial Properties ◽  
Topological Version ◽  
Aspherical Manifold

A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold from a given simplicial or cubical manifold. We investigate the combinatorics of several of these hyperbolizations and verify the Euler Characteristic Sign Conjecture for each of them. In addition, we explore further combinatorial properties of these hyperbolizations as they relate to several well-studied generating functions.

scholarly journals Sequence variations of the 1-2-3 conjecture and irregularity strength

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Ben Seamone ◽  
Brett Stevens
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Minimum Degree ◽  
List Size ◽  
Edge Weighting ◽  
Incident Edge ◽  
Sequence Variations ◽  
Distinct Sequence ◽  
International Audience ◽  
Irregularity Strength

Graph Theory International audience Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of E(G), and so two variations arise - one where we may choose any ordering of E(G) and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.

scholarly journals Some Results on the Eccentric Sequence of Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 55-57
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Degree Sequence ◽  
Integral Graph

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The degree of the vertex u is the number of vertices at distance one. The sequence of numbers of vertices having 0,1,2,3,... is called the degree sequence, which is the list of degrees of vertices of G arranged in non-decreasing order. The eccentricity e(a) of a is the distance of a farthest vertex from a. Let G be a connected graph. The Eccentric Sequence of G is the list of the eccentricities of its vertices arranged in non-decreasing order. In this paper, we characterize the eccentric sequence of some of the derived graphs namely the line graph of the integral graph , the eccentric sequence of Mycieleskian of a graph.

scholarly journals Functional Correctness of C Implementations of Dijkstra’s, Kruskal’s, and Prim’s Algorithms

2021 ◽  
pp. 801-826
Author(s):  
Anshuman Mohan ◽  
Wei Xiang Leow ◽  
Aquinas Hobor
Keyword(s):  
Graph Algorithms ◽  
Connected Graph ◽  
Spatial Representations ◽  
Undirected Graphs ◽  
Labeled Graphs ◽  
Standard Textbook ◽  
Spanning Forests ◽  
Common Notion ◽  
Functional Correctness ◽  
The Common

AbstractWe develop machine-checked verifications of the full functional correctness of C implementations of the eponymous graph algorithms of Dijkstra, Kruskal, and Prim. We extend Wang et al.’s CertiGraph platform to reason about labels on edges, undirected graphs, and common spatial representations of edge-labeled graphs such as adjacency matrices and edge lists. We certify binary heaps, including Floyd’s bottom-up heap construction, heapsort, and increase/decrease priority.Our verifications uncover subtle overflows implicit in standard textbook code, including a nontrivial bound on edge weights necessary to execute Dijkstra’s algorithm; we show that the intuitive guess fails and provide a workable refinement. We observe that the common notion that Prim’s algorithm requires a connected graph is wrong: we verify that a standard textbook implementation of Prim’s algorithm can compute minimum spanning forests without finding components first. Our verification of Kruskal’s algorithm reasons about two graphs simultaneously: the undirected graph undergoing MSF construction, and the directed graph representing the forest inside union-find. Our binary heap verification exposes precise bounds for the heap to operate correctly, avoids a subtle overflow error, and shows how to recycle keys to avoid overflow.

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