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.

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).

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.

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.

Some classifications of graphs with respect to a set adjacency relation

2020 ◽  
pp. 2050089
Author(s):  
G. Chiaselotti ◽  
T. Gentile ◽  
F. G. Infusino
Keyword(s):  
Computational Complexity ◽  
Tensor Product ◽  
Undirected Graph ◽  
Complete Graphs ◽  
Adjacency Relation ◽  
Vertex Set ◽  
Complete Bipartite Graphs ◽  
Graph Families ◽  
Graded Lattice ◽  
Product Of Graphs

For any finite simple undirected graph [Formula: see text], we consider the binary relation [Formula: see text] on the powerset [Formula: see text] of its vertex set given by [Formula: see text] if [Formula: see text], where [Formula: see text] denotes the neighborhood of a vertex [Formula: see text]. We call the above relation set adiacence dependency (sa)-dependency of [Formula: see text]. With the relation [Formula: see text] we associate an intersection-closed family [Formula: see text] of vertex subsets and the corresponding induced lattice [Formula: see text], which we call sa-lattice of [Formula: see text]. Through the equality of sa-lattices, we introduce an equivalence relation [Formula: see text] between graphs and propose three different classifications of graphs based on such a relation. Furthermore, we determine the sa-lattice for various graph families, such as complete graphs, complete bipartite graphs, cycles and paths and, next, we study such a lattice in relation to the Cartesian and the tensor product of graphs, verifying that in most cases it is a graded lattice. Finally, we provide two algorithms, namely, the T-DI ALGORITHM and the O-F ALGORITHM, in order to provide two different computational ways to construct the sa-lattice of a graph. For the O-F ALGORITHM we also determine its computational complexity.

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.

Graph-Theoretical Indices based on Simple, General and Complete Graphs

2013 ◽  
pp. 11-26
Author(s):  
Lionello Pogliani
Keyword(s):  
Complete Graphs ◽  
Molecular Connectivity ◽  
Theoretical Concepts ◽  
General Chemical ◽  
Starting Point ◽  
Wide Range ◽  
Connectivity Indices ◽  
Core Electrons ◽  
Multiple Edges ◽  
Electrotopological State

Valence molecular connectivity indices are based on the concept of valence delta, d v, that can be derived from general chemical graphs or chemical pseudographs. A general graph or pseudograph has multiple edges and loops and can be used to encode, through the valence delta, chemical entities. Two graph-theoretical concepts derived from chemical pseudographs are the intrinsic (I) and the electrotopological state (E) values, which are the used to define the valence delta of the pseudoconnectivity indices, ?I,S. Complete graphs encode, through a new valence delta, the core electrons of any atoms in a molecule. The connectivity indices, either valence connectivity or pseudoconnectivity, are the starting point to develop the dual connectivity indices. The dual indices show that not only can they assume negative values but also cover a wide range of numerical values. The central parameter of the molecular connectivity theory, the valence delta, defines a completely new set of connectivity indices, which can be distinguished by their configuration and advantageously used to model different properties and activities of compounds.

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.

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.

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