scholarly journals Double Arithmetic Odd Decomposition [DAOD] of Some Complete 4-Partite Graphs

2019 ◽  
Vol 9 (2) ◽  
pp. 3902-3907
Keyword(s):  
Undirected Graph ◽  
Edge Disjoint ◽  
Multiple Edges

Let G be a finite, connected, undirected graph without loops or multiple edges. If G1 , G2 , . . . ,Gn are connected edge – disjoint subgraphs of G with E(G) = E(G1 )  E(G2 )  . . .  E(Gn), then { G1 , G2 , . . . , Gn} is said to be a decomposition of G. The concept of Arithmetic Odd Decomposition [AOD] was introduced by E. Ebin Raja Merly and N. Gnanadhas . A decomposition {G1 , G2 , . . . , Gn } G is said to be Arithmetic Decomposition if each Gi is connected and | E(Gi )| = a+ (i – 1) d , for 1  i  n and a, d  ℤ . When a =1 and d = 2, we call the Arithmetic Decomposition as Arithmetic Odd Decomposition . A decomposition { G1 , G3 , . . . , G2n-1} of G is said to be AOD if | E (Gi ) | = i ,  i = 1, 3, . . . , 2n-1. In this paper, we introduce a new concept called Double Arithmetic Odd Decomposition [DAOD]. A graph G is said to have Double Arithmetic Odd Decomposition [DAOD] if G can be decomposed into 2k subgraphs { 2G1 , 2G3 , . . . , 2G2k-1 } such that each Gi is connected and | E (Gi ) | = i ,  i = 1, 3, . . . , 2k-1. Also we investigate DAOD of some complete 4-partite graphs such as K2,2,2,m , K2,4,4,m and K1 ,2,4,m .

Local edge and local node connectivity in regular graphs

2003 ◽  
Vol 40 (1-2) ◽  
pp. 151-158
Author(s):  
O. Fülöp
Keyword(s):  
Undirected Graph ◽  
Simple Graph ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Connectivity ◽  
Edge Disjoint ◽  
Node Connectivity ◽  
Multiple Edges ◽  
Local Edge

W. Mader [5] proved that every undirected graph (multiple edges are allowed but loops not) contains adjacent nodes x and y joined by min (d(x),dG(y))G edge-disjoint paths and in every undirected simple graph there are two adjacent nodes x and y joined by min (d(x),dG(y) Ginternally node-disjoint paths. In general it is not possible to fix x (or y) arbitrarily. The purpose of this paper is to provide conditions for the existence of a node x in d-regular graphs such that for all y joined to x there are d pairwise edge-disjoint (node-disjoint) paths between x and y. We also examine the directed version in case of local edge connectivity.

On an Algorithm for Ordering of Graphs

1971 ◽  
Vol 14 (2) ◽  
pp. 221-224 ◽  
Author(s):  
Milan Sekanina
Keyword(s):  
Undirected Graph ◽  
Multiple Edges

Let (G, ρ) be a finite connected (undirected) graph without loops and multiple edges. So x, y being two elements of G (vertices of the graph (G, ρ)), 〈x, y〉 ∊ ρ means that x and y are connected by an edge. Two vertices x, y ∊ G have the distance μ(x, y) equal to n, if n is the smallest number with the following property: there exists a sequence x0, x1, …, xn of vertices such that x0 = x, xn = y and 〈xi-1, Xi〉 ∊ ρ for i = 1, …, n. If x ∊ G, we put μ(x, x) = 0.

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).

Magic Valuations of Finite Graphs

1970 ◽  
Vol 13 (4) ◽  
pp. 451-461 ◽  
Author(s):  
Anton Kotzig ◽  
Alexander Rosa
Keyword(s):  
Undirected Graph ◽  
Complete Graphs ◽  
Finite Graphs ◽  
Vertex Set ◽  
Multiple Edges

The purpose of this paper is to investigate for graphs the existence of certain valuations which have some "magic" property. The question about the existence of such valuations arises from the investigation of another kind of valuations which are introduced in [1] and are related to cyclic decompositions of complete graphs into isomorphic subgraphs.Throughout this paper the word graph will mean a finite undirected graph without loops or multiple edges having at least one edge. By G(m, n) we denote a graph having m vertices and n edges, by V(G) and E(G) the vertex-set and the edge-set of G, respectively. Both vertices and edges are called the elements of the graph.

An Edge but not Vertex Transitive Cubic Graph*

1968 ◽  
Vol 11 (4) ◽  
pp. 533-535 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Undirected Graph ◽  
The Other ◽  
Cubic Graph ◽  
Vertex Transitive ◽  
Multiple Edges ◽  
Edge Transitive

Let G be an undirected graph, without loops or multiple edges. An automorphism of G is a permutation of the vertices of G that preserves adjacency. G is vertex transitive if, given any two vertices of G, there is an automorphism of the graph that maps one to the other. Similarly, G is edge transitive if for any two edges (a, b) and (c, d) of G there exists an automorphism f of G such that {c, d} = {f(a), f(b)}. A graph is regular of degree d if each vertex belongs to exactly d edges.

scholarly journals On Super Mean Labeling for Total Graph of Path and Cycle

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Nur Inayah ◽  
I. Wayan Sudarsana ◽  
Selvy Musdalifah ◽  
Nurhasanah Daeng Mangesa
Keyword(s):  
Undirected Graph ◽  
Total Graph ◽  
Vertex Set ◽  
Multiple Edges

Let G(V,E) be a graph with the vertex set V and the edge set E, respectively. By a graph G=(V,E) we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of G is called order of G and it is denoted by p. Let G be a (p,q) graph. A super mean graph on G is an injection f:V→{1,2,3…,p+q} such that, for each edge e=uv in E labeled by f⁎e=fu+f(v)/2, the set fV∪{f⁎e:e∈E} forms 1,2,3,…,p+q. A graph which admits super mean labeling is called super mean graph. The total graph T(G) of G is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G. We have showed that graphs T(Pn) and TCn are super mean, where Pn is a path on n vertices and Cn is a cycle on n vertices.

Some Geometries Associated with Parabolic Representations of Groups of Lie Type

1976 ◽  
Vol 28 (5) ◽  
pp. 1021-1031 ◽  
Author(s):  
Bruce N. Cooperstein
Keyword(s):  
Undirected Graph ◽  
Permutation Representation ◽  
Representations Of Groups ◽  
Symmetric Orbit ◽  
Multiple Edges ◽  
Groups Of Lie Type

Suppose (P, △) is an undirected graph without loops or multiple edges. We will denote by △ (x) the vertices adjacent to x and . Let (G, P) be a transitive permutation representation of a group G in a, set P, and Δ be a non-trivial self-paired (i.e. symmetric) orbit for the action of G on P X P. We identify △ with the set of all two subsets ﹛x, y﹜ with (x, y) in △. Then we have a graph (P, Δ) with G ≦ Aut (P, △), transitive on both P and △.

scholarly journals INTERSECTION GRAPH OF A MODULE

2013 ◽  
Vol 12 (05) ◽  
pp. 1250218 ◽  
Author(s):  
ERGÜN YARANERI
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Undirected Graph ◽  
Domination Number ◽  
Intersection Graph ◽  
Combinatorial Properties ◽  
Algebraic Properties ◽  
Vertex Set ◽  
Multiple Edges

Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph [Formula: see text] of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of [Formula: see text] to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for [Formula: see text]. For instance, we find the domination number of [Formula: see text]. We also find the chromatic number of [Formula: see text] in some cases. Furthermore, we study cycles in [Formula: see text], and complete subgraphs in [Formula: see text] determining the structure of V for which [Formula: see text] is planar.

scholarly journals Four-cycle free graphs, height functions, the pivot property and entropy minimality

2016 ◽  
Vol 37 (4) ◽  
pp. 1102-1132 ◽  
Author(s):  
NISHANT CHANDGOTIA
Keyword(s):  
Finite Number ◽  
Undirected Graph ◽  
Universal Cover ◽  
Single Site ◽  
Height Functions ◽  
Shift Space ◽  
Multiple Edges

Fix $d\geq 2$. Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding ‘vertex’ shift, $\text{Hom}(\mathbb{Z}^{d},{\mathcal{H}})$, denoted by $X_{{\mathcal{H}}}$. In this paper, we focus on ${\mathcal{H}}$ which is ‘four-cycle free’. There are two main results of this paper. Firstly, that $X_{{\mathcal{H}}}$ has the pivot property, meaning that, for all distinct configurations $x,y\in X_{{\mathcal{H}}}$, which differ only at a finite number of sites, there is a sequence of configurations $x=x^{1},x^{2},\ldots ,x^{n}=y\in X_{{\mathcal{H}}}$ for which the successive configurations $x^{i},x^{i+1}$ differ exactly at a single site. Secondly, if ${\mathcal{H}}$ is connected ,then $X_{{\mathcal{H}}}$ is entropy minimal, meaning that every shift space strictly contained in $X_{{\mathcal{H}}}$ has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in $X_{{\mathcal{H}}}$ to the universal cover of ${\mathcal{H}}$ and the introduction of ‘height functions’ in this context.

Sign in / Sign up

Export Citation Format

Share Document