scholarly journals Transition Restricted Gray Codes

10.37236/1235 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Bette Bultena ◽  
Frank Ruskey
Keyword(s):  
Undirected Graph ◽  
Gray Code ◽  
Hamilton Cycle ◽  
Gray Codes ◽  
Hamilton Path ◽  
Vertex Set ◽  
G Code

A Gray code is a Hamilton path $H$ on the $n$-cube, $Q_n$. By labeling each edge of $Q_n$ with the dimension that changes between its incident vertices, a Gray code can be thought of as a sequence $H = t_1,t_2,\ldots,t_{N-1}$ (with $N = 2^n$ and each $t_i$ satisfying $1 \le t_i \le n$). The sequence $H$ defines an (undirected) graph of transitions, $G_H$, whose vertex set is $\{1,2,\ldots,n\}$ and whose edge set $E(G_H) = \{ [t_i,t_{i+1}] \mid 1 \le i \le N-1 \}$. A $G$-code is a Hamilton path $H$ whose graph of transitions is a subgraph of $G$; if $H$ is a Hamilton cycle then it is a cyclic $G$-code. The classic binary reflected Gray code is a cyclic $K_{1,n}$-code. We prove that every tree $T$ of diameter 4 has a $T$-code, and that no tree $T$ of diameter 3 has a $T$-code.

scholarly journals Properties of Gray and Binary Representations

2004 ◽  
Vol 12 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Jonathan Rowe ◽  
Darrell Whitley ◽  
Laura Barbulescu ◽  
Jean-Paul Watson
Keyword(s):  
Genetic Algorithms ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Search Space ◽  
Gray Code ◽  
Global Optimum ◽  
Gray Codes ◽  
Neighborhood Structure ◽  
Local Optima ◽  
Vertex Set

Representations are formalized as encodings that map the search space to the vertex set of a graph. We define the notion of bit equivalent encodings and show that for such encodings the corresponding Walsh coefficients are also conserved. We focus on Gray codes as particular types of encoding and present a review of properties related to the use of Gray codes. Gray codes are widely used in conjunction with genetic algorithms and bit-climbing algorithms for parameter optimization problems. We present new convergence proofs for a special class of unimodal functions; the proofs show that a steepest ascent bit climber using any reflected Gray code representation reaches the global optimum in a number of steps that is linear with respect to the encoding size. There are in fact many different Gray codes.Shifting is defined as a mechanism for dynamically switching from one Gray code representation to another in order to escape local optima. Theoretical results that substantially improve our understanding of the Gray codes and the shifting mechanism are presented. New proofs also shed light on the number of unique Gray code neighborhoods accessible via shifting and on how neighborhood structure changes during shifting. We show that shifting can improve the performance of both a local search algorithm as well as one of the best genetic algorithms currently available.

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.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

scholarly journals Permutational labelling of constant weight Gray codes

2002 ◽  
Vol 65 (3) ◽  
pp. 399-406
Author(s):  
Inessa Levi ◽  
Steve Seif
Keyword(s):  
Gray Code ◽  
Gray Codes ◽  
Constant Weight ◽  
Single Exception ◽  
Positive Integers

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets of Xn with |Ai ∩ Ai+1| = r − 1 and a list of distinct partitions such that πi is orthogonal to both Ai and Ai+1, and πi+1 = πiλi for a suitable permutation λi of Xn.

Proper connection of power graphs of finite groups

2020 ◽  
pp. 2150033
Author(s):  
Xuanlong Ma
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
The Other ◽  
Connection Number ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Connection Number ◽  
A Finite Group

Let [Formula: see text] be a finite group. The power graph of [Formula: see text] is the undirected graph whose vertex set is [Formula: see text], and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of [Formula: see text] is the subgraph of the power graph of [Formula: see text] obtained by deleting all edges [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are two distinct elements of [Formula: see text]. In this paper, we determine the proper connection number of the reduced power graph of [Formula: see text]. As an application, we also determine the proper connection number of the power graph of [Formula: see text].

Graphs with a locally linear group of automorphisms

1995 ◽  
Vol 118 (2) ◽  
pp. 191-206 ◽  
Author(s):  
V. I. Trofimov ◽  
R. M. Weiss
Keyword(s):  
Permutation Group ◽  
Linear Group ◽  
Undirected Graph ◽  
Group Of Automorphisms ◽  
Vertex Set ◽  
Image Position ◽  
Pointwise Stabilizer ◽  
Locally Linear

Let Γ be an undirected graph, V(Γ) the vertex set of Γ and G a subgroup of aut(Γ). For each vertex x ↦ V(Γ), let Γx denote the set of vertices adjacent to x in Γ and the permutation group induced on Γx. by the stabilizer Gx. For each i ≥ 1, will denote the pointwise stabilizer in Gx of the set of vertices at distance at most i from x in Γ. Letfor each i ≥ 1 and any set of vertices x, y, …, z of Γ. An s-path (or s-arc) is an (s + 1)-tuple (x0, x1, … xs) of vertices such that xi ↦ Γxi–1 for 1 ≤ i ≤ s and xi ╪ xi–2 for 2 ≤ i ≤ s.

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.

Complementary equitably totally disconnected equitable domination in graphs

2020 ◽  
pp. 2150043
Author(s):  
P. Nataraj ◽  
R. Sundareswaran ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Totally Disconnected

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

Cycles and Connectivity in Graphs

1967 ◽  
Vol 19 ◽  
pp. 1319-1328 ◽  
Author(s):  
M. E. Watkins ◽  
D. M. Mesner
Keyword(s):  
Undirected Graph ◽  
Vertex Connectivity ◽  
Vertex Set ◽  
Multiple Edges

In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ = λ(G). One defines ζ to be the largest integer z not exceeding |V| such that for any set U ⊂ V with |U| = z, there is a cycle in G which contains U. The symbol i(U) will denote the component index of U. As a standard reference for this and other terminology, the authors recommend O. Ore (3).

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

2019 ◽  
Vol 28 (1) ◽  
pp. 109-119
Author(s):  
Ahmad Sabri ◽  
Vincent Vajnovszki
Keyword(s):  
Positive Integer ◽  
Gray Code ◽  
Lexicographic Order ◽  
Gray Codes ◽  
Fixed Length ◽  
Generating Algorithms

Abstract A generalized (resp. p-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than p) where in any of its prefixes each positive integer i occurs at most as often as any integer less than i. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and p-ary ballot sequences when p is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.

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