Weakly connected 2-domination in the join of graphs

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

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HF-Index and Y-Index of Some New Graph of Operations

International Journal of Engineering and Advanced Technology - Regular Issue ◽

10.35940/ijeat.c3351.0211322 ◽

2022 ◽

pp. 28-33

Author(s):

Dr. S. Nagarajan ◽

◽

G. Kayalvizhi ◽

G. Priyadharsini ◽

◽

...

Keyword(s):

Cartesian Product ◽

Graph Operations ◽

Join Of Graphs ◽

Product Of Graphs ◽

Corona Product

In this paper we derive HF index of some graph operations containing join, Cartesian Product, Corona Product of graphs and compute the Y index of new operations of graphs related to the join of graphs.

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The weakly connected independent set polytope in corona and join of graphs

Journal of Combinatorial Optimization ◽

10.1007/s10878-018-0275-9 ◽

2018 ◽

Vol 36 (3) ◽

pp. 1007-1023

Author(s):

F. Bendali ◽

J. Mailfert

Keyword(s):

Independent Set ◽

Join Of Graphs

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The spectra of a new join of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383091850074x ◽

2018 ◽

Vol 10 (06) ◽

pp. 1850074 ◽

Cited By ~ 2

Author(s):

Somnath Paul

Keyword(s):

Laplacian Spectrum ◽

Cospectral Graphs ◽

Signless Laplacian Spectrum ◽

Signless Laplacian ◽

Join Of Graphs ◽

Disjoint Sets

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

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Sandpile Groups and the Join of Graphs

Journal of Mathematical Sciences ◽

10.1007/s10958-013-1650-9 ◽

2014 ◽

Vol 196 (2) ◽

pp. 184-186

Author(s):

I. A Krepkiy

Keyword(s):

Join Of Graphs

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On the Roman {2}-domatic number of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092150052x ◽

2020 ◽

pp. 2150052

Author(s):

A. Giahtazeh ◽

H. R. Maimani ◽

A. Iranmanesh

Keyword(s):

Bipartite Graphs ◽

Domatic Number ◽

Join Of Graphs ◽

Complete Bipartite Graphs ◽

Appl Math ◽

Dominating Function

Let [Formula: see text] be a graph. A Roman[Formula: see text]-dominating function [Formula: see text] has the property that for every vertex [Formula: see text] with [Formula: see text], either [Formula: see text] is adjacent to a vertex assigned [Formula: see text] under [Formula: see text], or [Formula: see text] is adjacent to at least two vertices assigned [Formula: see text] under [Formula: see text]. A set [Formula: see text] of distinct Roman [Formula: see text]-dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a Roman[Formula: see text]-domination family (or functions) on [Formula: see text]. The maximum number of functions in a Roman [Formula: see text]-dominating family on [Formula: see text] is the Roman[Formula: see text]-domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we answer two conjectures of Volkman [L. Volkmann, The Roman [Formula: see text]-domatic number of graphs, Discrete Appl. Math. 258 (2019) 235–241] about Roman [Formula: see text]-domatic number of graphs and we study this parameter for join of graphs and complete bipartite graphs.

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The Generalized Distance Spectrum of the Join of Graphs

Symmetry ◽

10.3390/sym12010169 ◽

2020 ◽

Vol 12 (1) ◽

pp. 169 ◽

Cited By ~ 3

Author(s):

Abdollah Alhevaz ◽

Maryam Baghipur ◽

Hilal A. Ganie ◽

Yilun Shang

Keyword(s):

Spectral Properties ◽

Convex Combination ◽

Distance Matrix ◽

Regular Graphs ◽

Distance Spectrum ◽

Join Of Graphs ◽

Graph Operation ◽

Simple Connected Graph ◽

Symmetric Distance ◽

Generalized Distance

Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix D ( G ) and diagonal matrix of the vertex transmissions T r ( G ) . We determine the spectrum of the join of two graphs and of the join of a regular graph with another graph, which is the union of two different regular graphs. Moreover, thanks to the symmetry of the matrices involved, we study the generalized distance spectrum of the graphs obtained by generalization of the join graph operation through their eigenvalues of adjacency matrices and some auxiliary matrices.

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