The Zariski topology graph on scheme

2018 ◽  
Vol 13 (04) ◽  
pp. 2050075
Author(s):  
Mohamed Aqalmoun
Keyword(s):  
Nonempty Subset ◽  
Undirected Graph ◽  
Zariski Topology ◽  
Complement Graph ◽  
Irreducible Components ◽  
Vertex Set ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
The Relationship ◽  
Topology Graph

Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.

THE EXISTENCE OR NONEXISTENCE OF NON-COMMUTING GRAPHS WITH PARTICULAR PROPERTIES

2013 ◽  
Vol 13 (01) ◽  
pp. 1350064 ◽  
Author(s):  
M. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Strongly Regular Graph ◽  
Complete Multipartite Graph ◽  
Commuting Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Strongly Regular ◽  
Multipartite Graph ◽  
Complete Multipartite

We consider the non-commuting graph ∇(G) of a non-abelian finite group G; its vertex set is G\Z(G), the set of non-central elements of G, and two distinct vertices x and y are joined by an edge if [x, y] ≠ 1. We determine the structure of any finite non-abelian group G (up to isomorphism) for which ∇(G) is a complete multipartite graph (see Propositions 3 and 4). It is also shown that a non-commuting graph is a strongly regular graph if and only if it is a complete multipartite graph. Finally, it is proved that there is no non-abelian group whose non-commuting graph is self-complementary and n-cube.

scholarly journals Sharply Transitive $1$-Factorizations of Complete Multipartite Graphs

10.37236/349 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Giuseppe Mazzuoccolo ◽  
Gloria Rinaldi
Keyword(s):  
Complete Multipartite Graph ◽  
Transitive Action ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Even Order ◽  
Multipartite Graph ◽  
A Finite Group ◽  
Complete Multipartite ◽  
Sharply Transitive

Given a finite group $G$ of even order, which graphs $\Gamma$ have a $1$-factorization admitting $G$ as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when $\Gamma$ is a complete multipartite graph and $G$ is cyclic.

EXTREMAL CAYLEY GRAPHS OF FINITE CYCLIC GROUPS

2008 ◽  
Vol 09 (01n02) ◽  
pp. 73-82
Author(s):  
JOSEPH J. LEE ◽  
ELYSIA J. SHEU ◽  
XINGDE JIA
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Nonempty Subset ◽  
Undirected Graph ◽  
Average Distance ◽  
Additive Group ◽  
Cyclic Groups ◽  
Residue Classes ◽  
Vertex Set ◽  
A Finite Group

Let Γ be a finite group with a nonempty subset A. The Cayley graph Cay (Γ, A) of Γ generated by A is defined as the digraph with vertex set Γ and edge set {(x,y) | x-1 y ∈ A}. Cay (Γ, A) can be regarded as an undirected graph if x-1 ∈ A for all x ∈ A. Let [Formula: see text] denote the largest integer M so that there exists a set of integers A = {±1, ±a2;…, ±ak} such that the average distance between all pairs of vertices of Cay (ℤM,A) is at most r, where ℤM is the additive group of residue classes modulo M. It is proved in this paper that [Formula: see text] It is also proved that [Formula: see text]

scholarly journals On groups admitting no integral Cayley graphs besides complete multipartite graphs

2013 ◽  
Vol 7 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Alireza Abdollahi ◽  
Mojtaba Jazaeri
Keyword(s):  
Cayley Graph ◽  
Identity Element ◽  
Simple Groups ◽  
Complete Multipartite Graph ◽  
Arbitrary Graph ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Vertex Set ◽  
Multipartite Graph ◽  
Complete Multipartite

Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.

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 k-Tuple Total Domination in Complementary Prisms

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Adel P. Kazemi
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Complementary Prism

Let k be a positive integer, and let G be a graph with minimum degree at least k. In their study (2010), Henning and Kazemi defined the k-tuple total domination number γ×k,tG of G as the minimum cardinality of a k-tuple total dominating set of G, which is a vertex set such that every vertex of G is adjacent to at least k vertices in it. If G̅ is the complement of G, the complementary prism GG̅ of G is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. In this paper, we extend some of the results of Haynes et al. (2009) for the k-tuple total domination number and also obtain some other new results. Also we find the k-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.

scholarly journals Computing metric dimension of compressed zero divisor graphs associated to rings

2018 ◽  
Vol 10 (2) ◽  
pp. 298-318
Author(s):  
S. Pirzada ◽  
M. Imran Bhat
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
The Family ◽  
Vertex Set ◽  
Relationship Of ◽  
The Relationship

Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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