scholarly journals Uniformly Most Reliable Three-Terminal Graph of Dense Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sun Xie ◽  
Haixing Zhao ◽  
Jun Yin
Keyword(s):  
Complete Graph ◽  
Connection Probability ◽  
Dense Graphs ◽  
Vertex Set ◽  
Terminal Reliability ◽  
Terminal Graph

A graph G with k specified target vertices in vertex set is a k -terminal graph. The k -terminal reliability is the connection probability of the fixed k target vertices in a k -terminal graph when every edge of this graph survives independently with probability p . For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves, and Sun constructed a locally most reliable two-terminal graph for p close to 1 and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with n vertices and m edges. First, we give the locally most reliable three-terminal graphs of n and m in certain ranges for p close to 0 and 1. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific n and m , where n ≥ 7 and n 2 − ⌊ n − 3 / 2 ⌋ ≤ m ≤ n 2 − 2 . Finally, some uniformly most reliable graphs are given for n vertices and m edges, where 4 ≤ n ≤ 6 and m = n 2 − 2 or n ≥ 5 and m = n 2 − 1 .

Characterizing 2-distance graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Ramuel P. Ching ◽  
I. J. L. Garces
Keyword(s):  
Complete Graph ◽  
Distance Graph ◽  
Simple Graph ◽  
Distance Graphs ◽  
Vertex Set

Let [Formula: see text] be a finite simple graph. The [Formula: see text]-distance graph [Formula: see text] of [Formula: see text] is the graph with the same vertex set as [Formula: see text] and two vertices are adjacent if and only if their distance in [Formula: see text] is exactly [Formula: see text]. A graph [Formula: see text] is a [Formula: see text]-distance graph if there exists a graph [Formula: see text] such that [Formula: see text]. In this paper, we give three characterizations of [Formula: see text]-distance graphs, and find all graphs [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] is an integer, [Formula: see text] is the path of order [Formula: see text], and [Formula: see text] is the complete graph of order [Formula: see text].

scholarly journals Complete graph immersions in dense graphs

2017 ◽  
Vol 340 (5) ◽  
pp. 1019-1027 ◽  
Author(s):  
Sylvia Vergara S.
Keyword(s):  
Complete Graph ◽  
Dense Graphs

scholarly journals Upper density of monochromatic infinite paths

2019 ◽  
Author(s):  
Jan Corsten ◽  
Louis DeBiasio ◽  
Ander Lamaison ◽  
Richard Lang
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
Classical Case ◽  
Infinite Graphs ◽  
Edge Colorings ◽  
Vertex Set ◽  
Upper Density ◽  
Countably Infinite ◽  
Positive Integers ◽  
Red Path

Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.

scholarly journals Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

10.37236/773 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jeremy F. Alm ◽  
Roger D. Maddux ◽  
Jacob Manske
Keyword(s):  
Complete Graph ◽  
Ramsey Theory ◽  
Projective Planes ◽  
Relation Algebras ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
Finite Set ◽  
Vertex Set ◽  
Special Case

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.

CONSTRUCTING MULTIDIMENSIONAL SPANNER GRAPHS

1991 ◽  
Vol 01 (02) ◽  
pp. 99-107 ◽  
Author(s):  
JEFFERY S. SALOWE
Keyword(s):  
Shortest Path ◽  
Complete Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Input Parameter ◽  
Minimum Spanning Trees ◽  
Positive Edge ◽  
Spanning Subgraph ◽  
Vertex Set

Given a connected graph G=(V,E) with positive edge weights, define the distance dG(u,v) between vertices u and v to be the length of a shortest path from u to v in G. A spanning subgraph G' of G is said to be a t-spanner for G if, for every pair of vertices u and v, dG'(u,v)≤t·dG(u,v). Consider a complete graph G whose vertex set is a set of n points in [Formula: see text] and whose edge weights are given by the Lp distance between respective points. Given input parameter ∊, 0<∊≤1, we show how to construct a (1+∊)-spanner for G containing [Formula: see text] edges in [Formula: see text] time. We apply this spanner to the construction of approximate minimum spanning trees.

Automorphisms of the co-maximal ideal graph over matrix ring

2017 ◽  
Vol 16 (12) ◽  
pp. 1750226 ◽  
Author(s):  
Dengyin Wang ◽  
Li Chen ◽  
Fenglei Tian
Keyword(s):  
Finite Field ◽  
Complete Graph ◽  
Maximal Ideal ◽  
Matrix Ring ◽  
Invertible Matrix ◽  
Vertex Set ◽  
Matrix Formula ◽  
Field Automorphism

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] be the ring of all [Formula: see text] matrices over [Formula: see text], [Formula: see text] be the set of all nontrivial left ideals of [Formula: see text]. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph with [Formula: see text] as vertex set and two nontrivial left ideals [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. If [Formula: see text], it is easy to see that [Formula: see text] is a complete graph, thus any permutation of vertices of [Formula: see text] is an automorphism of [Formula: see text]. A natural problem is: How about the automorphisms of [Formula: see text] when [Formula: see text]. In this paper, we aim to solve this problem. When [Formula: see text], a mapping [Formula: see text] on [Formula: see text] is proved to be an automorphism of [Formula: see text] if and only if there is an invertible matrix [Formula: see text] and a field automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text].

Topological covers of complete graphs

1998 ◽  
Vol 123 (3) ◽  
pp. 549-559 ◽  
Author(s):  
A. GARDINER ◽  
CHERYL E. PRAEGER
Keyword(s):  
Complete Graph ◽  
Additional Assumption ◽  
Complete Graphs ◽  
Symmetric Graph ◽  
Quotient Graph ◽  
Vertex Set ◽  
Acts 2

Let Γ be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition [Bscr ], with blocks B∈[Bscr ] of size v and with k[les ]v independent edges joining each pair of adjacent blocks. In a previous paper we introduced a framework for analysing such graphs Γ in terms of (a) the natural quotient graph Γ[Bscr ] of valency b=vr/k, and (b) the 1-design [Dscr ](B) induced on each block. Here we examine the case where k=v and Γ[Bscr ]=Kb+1 is a complete graph. The 1-design [Dscr ](B) is then degenerate, so gives no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove that there is then a unique exceptional graph for which [mid ]B[mid ]=v>b+1.

scholarly journals On integral sum labeling of dense graphs

2010 ◽  
Vol 41 (4) ◽  
pp. 317-324
Author(s):  
T. Nicholas
Keyword(s):  
Existence Theorem ◽  
Connected Graph ◽  
Alternative Proof ◽  
Dense Graphs ◽  
Vertex Set ◽  
The Difference ◽  
Positive Integers

A graph is said to be a \textit{sum graph} if there exists a set $S$ of positive integers as its vertex set with two vertices adjacent whenever their sum is in $S$. An integral sum graph is defined just as the sum graph, the difference being that the label set $S$ is a subset of $Z$ instead of set of positive integers. The sum number of a given graph $G$ is defined as the smallest number of isolated vertices which when added to $G$ results in a sum graph. The integral sum number of $G$ is analogous. In this paper, we mainly prove that any connected graph $G$ of order $n$ with at least three vertices of degree $(n-1)$ is not an integral sum graph. We characterise the integral sum graph $G$ of order $n$ having exactly two vertices of degree $(n-1)$ each and hence give an alternative proof for the existence theorem of sum graphs.

Bounds on the sum of domination number and metric dimension of graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Cong X. Kang ◽  
Eunjeong Yi
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Domination Number ◽  
Unicyclic Graph ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
Dimension Formula ◽  
Vertex Set ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text]. The domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set [Formula: see text] such that every vertex not in [Formula: see text] is adjacent to a vertex in [Formula: see text]. The metric dimension, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set of vertices such that every vertex of [Formula: see text] is uniquely determined by its vector of distances to the chosen vertices. For a tree [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text] denotes the number of exterior major vertices of [Formula: see text]; further, we characterize trees [Formula: see text] achieving equality. For a connected graph [Formula: see text] of order [Formula: see text], Bagheri Gh. et al. proved that [Formula: see text] and equality holds if and only if [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the complete graph and [Formula: see text] denotes a complete bi-partite graph of order [Formula: see text]. We characterize graphs [Formula: see text] for which [Formula: see text] equals two and three, respectively. We also characterize graphs [Formula: see text] satisfying [Formula: see text] when [Formula: see text] is a tree, a unicyclic graph, or a complete multi-partite graph.

Sign in / Sign up

Export Citation Format

Share Document