scholarly journals Random Cayley digraphs of diameter 2 and given degree

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Manuel E. Lladser ◽  
Primož Potočnik ◽  
Jozef Širáň ◽  
Mark C. Wilson
Keyword(s):  
Graph Theory ◽  
Convergence Rate ◽  
Abelian Groups ◽  
Cayley Digraph ◽  
Generating Sets ◽  
International Audience

Graph Theory International audience We consider random Cayley digraphs of order n with uniformly distributed generating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n -> infinity and k = f(n), focusing on the functions f(n) = left perpendicularn(delta)right perpendicular and f(n) = left perpendicularcnright perpendicular. In both instances we show that this probability converges to 1 as n -> infinity for arbitrary fixed delta is an element of (1/2, 1) and c is an element of (0, 1/2), respectively, with a much larger convergence rate in the second case and with sharper results for Abelian groups.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals The resolving number of a graph Delia

2013 ◽  
Vol Vol. 15 no. 3 (Graph Theory) ◽  
Author(s):  
Delia Garijo ◽  
Antonio González ◽  
Alberto Márquez
Keyword(s):  
Graph Theory ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Clique Number ◽  
Metric Dimension ◽  
Np Hard ◽  
Arbitrary Graph ◽  
International Audience ◽  
Graph Parameters ◽  
Two Parameters

Graph Theory International audience We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals Guarded subgraphs and the domination game

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Boštjan Brešar ◽  
Sandi Klavžar ◽  
Gasper Košmrlj ◽  
Doug F. Rall
Keyword(s):  
Graph Theory ◽  
Domination Number ◽  
Metric Properties ◽  
International Audience ◽  
Game Domination Number ◽  
Domination Game

Graph Theory International audience We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

scholarly journals The total irregularity of a graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Hosam Abdo ◽  
Stephan Brandt ◽  
D. Dimitrov
Keyword(s):  
Graph Theory ◽  
Degree Of A Vertex ◽  
International Audience

Graph Theory International audience In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.

scholarly journals On the Meyniel condition for hamiltonicity in bipartite digraphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Janusz Adamus ◽  
Lech Adamus ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Minimal Degree ◽  
Sufficient Condition ◽  
Type Criterion ◽  
Bipartite Digraph ◽  
Strongly Connected ◽  
International Audience

Graph Theory International audience We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a&#x2265;2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u)+d(v)&#x2265;3a whenever uv∉A(D) and vu∉A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if δ(D)&#x2265;3a/2.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

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