scholarly journals Decomposition of cartesian product of complete graphs into sunlet graphs of order eight

2022 ◽  
pp. 27-43
Author(s):  
Kaliappan SOWNDHARİYA ◽  
Appu MUTHUSAMY
Keyword(s):  
Cartesian Product ◽  
Complete Graphs

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 Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

scholarly journals Grundy Number of Some Chordal Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 64
Author(s):  
R. Nagarathinam ◽  
N. Parvathi ◽  
. .
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
Chordal Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Grundy Number

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c : V → {1, 2, . . .} such that c(u) 12≠"> c(v) for every edge uv ∈ E. For proper coloring, colors assigned must be minimum, but for Grundy coloring which should be maximum. In this instance, Grundy numbers of chordal graphs like Cartesian product of two path graphs, join of the path and complete graphs and the line graph of tadpole have been executed 

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

scholarly journals Constructing Node-Disjoint Routes in K-Ary N-Cubes

1997 ◽  
Vol 3 ◽  
pp. 41
Author(s):  
Khaled Day ◽  
Abdel Elah Al-Ayyoub
Keyword(s):  
General Result ◽  
Interconnection Networks ◽  
Minimum Distance ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Factor Graphs ◽  
Complete Set ◽  
Alternate Routes

In this paper, a method for constructing node-disjoint (parallel) paths in k-ary n-cube interconnection networks is described. We start by showing in general how to construct parallel paths in any Cartesian product of two graphs based on known paths in the factor graphs. Then we apply the general result to build a complete set of parallel paths (i.e., as many paths as the degree of the network) between any two nodes of a k-ary n-cube which can be viewed as the Cartesian product of complete graphs. Each of the constructed paths is of length at most 2 plus the minimum distance between the two nodes. These parallel paths are useful in speeding-up the transfer of large amounts of data between two nodes and in offering alternate routes in cases of faulty nodes.

scholarly journals On the Spanning Trees of the Hypercube and other Products of Graphs

10.37236/2510 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Olivier Bernardi
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Cartesian Product ◽  
Product Formula ◽  
General Context ◽  
Complete Graphs ◽  
Independence Property ◽  
Number Of Spanning Trees ◽  
Tree Approach

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

scholarly journals The crossing number of the Cartesian product of paths with complete graphs

2014 ◽  
Vol 328 ◽  
pp. 71-78 ◽  
Author(s):  
ZhangDong Ouyang ◽  
Jing Wang ◽  
YuanQiu Huang
Keyword(s):  
Cartesian Product ◽  
Crossing Number ◽  
Complete Graphs
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