K − Bipartite Matching Extendability of Special Graphs

2011 ◽  
Vol 121-126 ◽  
pp. 4008-4012
Author(s):  
Zhi Hao Hui ◽  
Jin Wei Yang ◽  
Biao Zhao
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Perfect Matching ◽  
Bipartite Matching ◽  
Simple Connected Graph ◽  
Matching Extendability

Let be a simple connected graph containing a perfect matching. For a positive integer , , is said to be bipartite matching extendable if every bipartite matching of with is included in a perfect matching of . In this paper, we show that bipartite matching extendability of some special graphs.

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

scholarly journals Chromatic Zagreb indices for graphical embodiment of colour clusters

2019 ◽  
Vol 3 (1) ◽  
pp. 48
Author(s):  
Johan Kok ◽  
Sudev Naduvath ◽  
Muhammad Kamran Jamil
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Chromatic Numbers ◽  
Zagreb Indices ◽  
Colour Class ◽  
Graph Structures ◽  
Simple Connected Graph

<p>For a colour cluster <span class="math"><em>C</em> = (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, …, C<sub>ℓ</sub>)</span>, where <span class="math">C<sub><em>i</em></sub></span> is a colour class such that <span class="math">∣C<sub><em>i</em></sub>∣ = <em>r</em><sub><em>i</em></sub></span>, a positive integer, we investigate two types of simple connected graph structures <span class="math"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class="math"><em>G</em><sub>2</sub><sup><em>C</em></sup></span> which represent graphical embodiments of the colour cluster such that the chromatic numbers <span class="math"><em>χ</em>(<em>G</em><sub>1</sub><sup><em>C</em></sup>) = <em>χ</em>(<em>G</em><sub>2</sub><sup><em>C</em></sup>) = ℓ</span> and <span class="math">$\min\{\varepsilon(G^{C}_1)\}=\min\{\varepsilon(G^{C}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$</span>, and <span class="math"><em>ɛ</em>(<em>G</em>)</span> is the size of a graph <span class="math"><em>G</em></span>. In this paper, we also discuss the chromatic Zagreb indices corresponding to <span class="math"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class="math"><em>G</em><sub>2</sub><sup><em>C</em></sup></span>.</p>

scholarly journals Matching Extendabilities of G = Cm ∨ Pn

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 941 ◽  
Author(s):  
Zhi-hao Hui ◽  
Yu Yang ◽  
Hua Wang ◽  
Xiao-jun Sun
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bipartite Matching ◽  
Induced Matching ◽  
Necessary And Sufficient ◽  
Graph Join ◽  
Matching Extendability

A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G = C m ∨ P n ( m ≥ 3 and n ≥ 1 ) be the graph join of C m (the cycle with m vertices) and P n (the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable.

BIPARTITE MATCHING EXTENDABILITY AND TOUGHNESS

2010 ◽  
Vol 02 (01) ◽  
pp. 33-44
Author(s):  
XIUMEI WANG ◽  
SUJING ZHOU ◽  
YIXUN LIN
Keyword(s):  
Perfect Matching ◽  
Necessary Conditions ◽  
Simple Graph ◽  
Sufficient Condition ◽  
Bipartite Matching ◽  
Bipartite Subgraph ◽  
Matching Extendability

Let G be a simple graph containing a perfect matching. G is said to be bipartite matching extendable (BM-extendable) if every matching M which is a perfect matching of an induced bipartite subgraph extends to a perfect matching of G. In this paper, we study some relations between toughness and BM-extendability of a graph, including some sufficient or necessary conditions about toughness for a graph to be BM-extendable, and a sufficient condition for a BM-extendable graph to be 1-tough.

scholarly journals On the average of the eccentricities of a graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1395-1401 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Xia Li ◽  
Haiqiong Liu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Mean Value ◽  
Extremal Graphs ◽  
The Mean ◽  
Average Eccentricity ◽  
General Graphs ◽  
Simple Connected Graph

Let G = (V; E) be a simple connected graph of order n with m edges. Also let eG(vi) be the eccentricity of a vertex vi in G. We can assume that eG(v1) eG(v2) ? ... ? eG(vn-1) ? eG(vn). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, avec(G) = 1/n ?n,i=1 eG(vi). Let ? = ?G be the largest positive integer such that eG(vG ) ? avec(G). In this paper, we study the value of G of a graph G. For any tree T of order n, we prove that 2 ? ?T ? n - 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n,2 ? ?G ? n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on ?G of general graphs G.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

2007 ◽  
Vol 3 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Degree Of Vertex ◽  
Simple Connected Graph ◽  
Ga Index

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)".

scholarly journals On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

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