Divide-and-Conquer based All Spanning Tree Generation Algorithm of a Simple Connected Graph

2021 ◽  
Author(s):  
Maumita Chakraborty ◽  
Ranjan Mehera ◽  
Rajat Kumar Pal
Keyword(s):  
Connected Graph ◽  
Spanning Tree ◽  
Divide And Conquer ◽  
Generation Algorithm ◽  
Tree Generation ◽  
Simple Connected Graph

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.

An Improved Test Tree Generation Algorithm from a Graphical Model

2021 ◽  
Author(s):  
Pan Liu ◽  
Yihao Li ◽  
Hao Chen ◽  
Xuankui Zheng ◽  
Sihao Huang
Keyword(s):  
Graphical Model ◽  
Generation Algorithm ◽  
Tree Generation ◽  
Test Tree

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 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.

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

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 The edge geodetic self decomposition number of a graph

2020 ◽  
Author(s):  
John J ◽  
Stalin D
Keyword(s):  
Connected Graph ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Edge Disjoint ◽  
General Properties ◽  
Geodetic Number ◽  
Simple Connected Graph ◽  
Decomposition Number

Let  G = (V, E)  be a simple connected  graph  of order  p and  size q.  A decomposition  of a graph  G is a collection  π  of edge-disjoint sub graphs  G1, G2, ..., Gn  of G such  that every  edge of G belongs to exactly  one Gi , (1 ≤ i ≤ n) . The decomposition  π = {G1, G2, ....Gn } of a connected  graph  G is said to be an edge geodetic self decomposi- tion  if ge (Gi ) = ge (G), (1 ≤ i ≤ n).The maximum  cardinality of π is called the edge geodetic self decomposition  number of G and is denoted by πsge (G), where ge (G) is the edge geodetic number  of G.  Some general properties   satisfied  by  this  concept  are  studied.    Connected  graphs which are edge geodetic self decomposable  are characterized.

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