A Criterion for (Non-)Planarity of The Block-Transformation Graph Gαβγ when αβγ = 101

2014 ◽  
Vol 10 ◽  
pp. 38-47 ◽  
Author(s):  
B. Basavanagoud ◽  
Jaishri B. Veeragoudar
Keyword(s):  
General Concept ◽  
Crossing Number ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
Necessary And Sufficient

The general concept of the block-transformation graph Gαβγ was introduced in [1]. The vertices and blocks of a graph are its members. The block-transformation graph G101 of a graph G is the graph, whose vertex set is the union of vertices and blocks of G, in which two vertices are adjacent whenever the corresponding vertices of G are adjacent or the corresponding blocks of G are nonadjacent or the corresponding members of G are incident. In this paper, we present characterizations of graphs whose block-transformation graphs G101 are planar, outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the block-transformation graph G101 to have crossing number one.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

scholarly journals Qlick Graphs with Crossing Number One

2016 ◽  
Vol 17 ◽  
pp. 75-84
Author(s):  
Bommanahal Basavanagoud ◽  
V.R. Kulli
Keyword(s):  
Crossing Number ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Forbidden Subgraphs ◽  
Necessary And Sufficient

In this paper, we deduce a necessary and sufficient condition for graphs whose qlick graphs have crossing number one. We also obtain a necessary and sufficient condition for qlick graphs to have crossing number one in terms of forbidden subgraphs.

scholarly journals Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

10.37236/2390 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Xiaogang Liu ◽  
Sanming Zhou
Keyword(s):  
Spectral Properties ◽  
Line Graph ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absolute Value ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Necessary And Sufficient

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

scholarly journals Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Shonda Gosselin ◽  
Andrzej Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
International Audience ◽  
Complete Multipartite ◽  
Necessary And Sufficient

Combinatorics International audience A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.

scholarly journals Semi-Image Neighborhood Block Graphs with Crossing Numbers

2013 ◽  
Vol 5 (2) ◽  
pp. 295-299
Author(s):  
K. M. Niranjan ◽  
P. Nagaraja ◽  
Lokesh V
Keyword(s):  
Graph Theory ◽  
Crossing Number ◽  
Sufficient Condition ◽  
Block Graph ◽  
Necessary And Sufficient Condition ◽  
Engineering Applications ◽  
Crossing Numbers ◽  
Block Graphs ◽  
Necessary And Sufficient

The advent of graph theory has played a prominent role in wide variety of engineering applications. Minimization of crossing numbers in graphs optimizes its use in many applications. In this paper, we establish the necessary and sufficient condition for Semi-Image neighborhood block graph to have crossing number 3. We also prove that the Semi-Image neighborhood block graph NBI(G) of a graph never has crossing numbers k, where    1 ? k ? 6.Keywords: Semi-Image; Neighborhood block; Crossing numbers.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i2.11019        J. Sci. Res. 5 (2), 295-299 (2013)

scholarly journals Dynamics of Boundary Graphs

2013 ◽  
Vol 5 (3) ◽  
pp. 447-455
Author(s):  
G. Mariumuthu ◽  
M. S. Saraswathy
Keyword(s):  
Simple Graph ◽  
The Other ◽  
Boundary Vertex ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Periodic Graph ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Boundary Graph

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if for all The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the mth iterated boundary graph of G is defined as A graph G is periodic if for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that We give the necessary and sufficient condition for a graph to be eventually periodic.  Keywords: Boundary graph; Periodic graph. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.14866 J. Sci. Res. 5 (3), xxx-xxx (2013) 

scholarly journals Граф блоков

2019 ◽  
Author(s):  
A. Kelkar ◽  
K. Jaysurya ◽  
H.M. Nagesh
Keyword(s):  
Crossing Number ◽  
Sufficient Condition ◽  
Block Graph ◽  
Necessary And Sufficient Condition ◽  
Cut Vertex ◽  
Necessary And Sufficient

The block graph of a graph $G$, written $B(G)$, is the graph whose vertices are the blocks of $G$ and in which two vertices are adjacent whenever the corresponding blocks have a cut-vertex in common. We study the properties of $B(G)$ and present the characterization of graphs whose $B(G)$ are planar, outerplanar, maximal outerplanar, minimally non-outerplanar, Eulerian, and Hamiltonian. A necessary and sufficient condition for $B(G)$ to have crossing number one is also presented.

Digraph from power mapping on noncommutative groups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050084
Author(s):  
Jinxing Zhao ◽  
Guixin Deng
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Cyclic Groups ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Power Mapping

Let [Formula: see text] be a group and [Formula: see text] be a positive integer. The [Formula: see text]-power digraph [Formula: see text] is consisting of vertex set [Formula: see text] and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text]. We study the [Formula: see text]-power digraph on the semiproduct of cyclic groups. In particular, we obtain the distribution of indegree and cycles, and determine the structure of trees attached with vertices of power digraph. Finally, we establish a necessary and sufficient condition for isomorphism of digraphs [Formula: see text] and [Formula: see text].

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