scholarly journals Notion of Valued Set

2021 ◽  
Author(s):  
Henry Garrett
Keyword(s):  
Vertex Set ◽  
Algebraic Operations ◽  
Valued Graph ◽  
General Graphs

The aim of this article is to introduce the new notion on a given graph. The notions of valued set, valued function, valued graph and valued quotient are introduced. The attributes of these new notions are studied. Valued set is about the set of vertices which have the maximum number of neighbors. The kind of partition of the vertex set to the vertices of the valued set is introduced and its attributes are studied. The behaviors of classes of graphs under these new notions are studied and the algebraic operations on these sets in the different situations get new result to understand the classes of graphs, these notions and the general graphs better and more.

A new graph radio k-coloring algorithm

2019 ◽  
Vol 11 (01) ◽  
pp. 1950005 ◽  
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi
Keyword(s):  
Connected Graph ◽  
Channel Assignment ◽  
Time Complexity ◽  
Radio Spectrum ◽  
Circulant Graph ◽  
Communication Services ◽  
Channel Assignment Problem ◽  
Vertex Set ◽  
General Graphs ◽  
Simple Connected Graph

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio [Formula: see text]-coloring of graphs is a variation of CAP. For a positive integer [Formula: see text], a radio [Formula: see text]-coloring of a simple connected graph [Formula: see text] is a mapping [Formula: see text] from the vertex set [Formula: see text] to the set [Formula: see text] of non-negative integers such that [Formula: see text] for each pair of distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of a radio [Formula: see text]-coloring [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] and the radio[Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text] where the minimum is taken over all radio [Formula: see text]-coloring of [Formula: see text]. In this paper, we present two radio [Formula: see text]-coloring algorithms for general graphs which will produce radio [Formula: see text]-colorings with their spans. For an [Formula: see text]-vertex simple connected graph the time complexity of the both algorithm is of [Formula: see text]. Implementing these algorithms we get the exact value of [Formula: see text] for several graphs (for example, [Formula: see text], [Formula: see text], [Formula: see text], some circulant graph etc.) and many values of [Formula: see text], especially for [Formula: see text].

General eccentric distance sum of graphs

2020 ◽  
pp. 2150046
Author(s):  
Tomáš Vetrík
Keyword(s):  
Connected Graph ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Vertex Set ◽  
General Graphs ◽  
Eccentric Distance

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

The restrained k-rainbow reinforcement number of graphs

2020 ◽  
pp. 2150026
Author(s):  
Saeed Shaebani ◽  
Saeed Kosari ◽  
Leila Asgharsharghi
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
General Graphs

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple graph. A restrained [Formula: see text]-rainbow dominating function (R[Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text], such that every vertex [Formula: see text] with [Formula: see text] satisfies both of the conditions [Formula: see text] and [Formula: see text] simultaneously, where [Formula: see text] denotes the open neighborhood of [Formula: see text]. The weight of an R[Formula: see text]RDF is the value [Formula: see text]. The restrained[Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an R[Formula: see text]RDF of [Formula: see text]. The restrained[Formula: see text]-rainbow reinforcement number [Formula: see text] of [Formula: see text], is defined to be the minimum number of edges that must be added to [Formula: see text] in order to decrease the restrained [Formula: see text]-rainbow domination number. In this paper, we determine the restrained [Formula: see text]-rainbow reinforcement number of some special classes of graphs. Also, we present some bounds on the restrained [Formula: see text]-rainbow reinforcement number of general graphs.

scholarly journals Spectra of Edge-Independent Random Graphs

10.37236/3576 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Linyuan Lu ◽  
Xing Peng
Keyword(s):  
Random Graphs ◽  
Adjacency Matrix ◽  
Error Term ◽  
Type Inequality ◽  
Laplacian Matrix ◽  
Mild Conditions ◽  
Multiplicative Factor ◽  
Indicator Variables ◽  
Vertex Set ◽  
General Graphs

Let $G$ be a random graph on the vertex set $\{1,2,\ldots, n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probabilities $p_{ij}$ for $\{i,j\}$ being an edge in $G$ are not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recently studied by Oliveira and Chung-Radcliffe. Let $A$ be the adjacency matrix of $G$, $\bar A={\rm E}(A)$, and $\Delta$ be the maximum expected degree of $G$.  Oliveira first proved that asymptotically almost surely $\|A-\bar A\|=O(\sqrt{\Delta \ln n})$ provided $\Delta\geq C \ln n$ for some constant $C$. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that asymptotically almost surely $\|A-\bar A\|\leq (2+o(1))\sqrt{\Delta}$ with a slightly stronger condition $\Delta\gg \ln^4 n$.  For the Laplacian matrix $L$ of $G$, Oliveira and Chung-Radcliffe proved similar results $\|L-\bar L\|=O(\sqrt{\ln n}/\sqrt{\delta})$ provided the minimum expected degree $\delta\geq C' \ln n$ for some constant $C'$; we also improve their results by removing the $\sqrt{\ln n}$ multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classical Erdős–Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.

THE SIGNED BAD NUMBERS IN GRAPHS

2011 ◽  
Vol 03 (01) ◽  
pp. 33-41 ◽  
Author(s):  
A. N. GHAMESHLOU ◽  
ABDOLLAH KHODKAR ◽  
S. M. SHEIKHOLESLAMI
Keyword(s):  
Complete Graph ◽  
Upper Bounds ◽  
Vertex Set ◽  
Closed Neighborhood ◽  
General Graphs

The closed neighborhood NG[v] of a vertex v in a graph G is the set consisting of v and of all neighborhood vertices of v. Let f be a function on V(G), the vertex set of G, into the set {-1, 1}. If ∑u∈N[v] f(u) ≤ 1 for all vertices v of G, then f is called a signed bad function of G. The maximum of the values of ∑v∈V(G) f(v), taking the maximum over all signed bad functions f of G, is called the signed bad number of G and denoted by β s (G). In this paper, we establish some upper bounds on the signed bad numbers for general graphs. In addition, we determine β s (G), when G is a complete graph, a cycle or a path.

scholarly journals Constructing universal graphs for induced-hereditary graph properties

2013 ◽  
Vol 63 (2) ◽  
Author(s):  
Izak Broere ◽  
Johannes Heidema ◽  
Peter Mihók
Keyword(s):  
Recursive Construction ◽  
Binary Expansion ◽  
Hereditary Property ◽  
Induced Subgraph ◽  
Finite Graphs ◽  
Graph Properties ◽  
Vertex Set ◽  
Universal Graphs ◽  
Positive Integers ◽  
General Graphs

AbstractRado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set I of all countable graphs (since every graph in I is isomorphic to an induced subgraph of R).In this paper we describe a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of a universal graph for the set of finite graphs in a product of properties of graphs is also presented.The paper is concluded by a comparison between the two types of construction of universal graphs.

scholarly journals Network topology inference using information cascades with limited statistical knowledge

2019 ◽  
Vol 9 (2) ◽  
pp. 327-360
Author(s):  
Feng Ji ◽  
Wenchang Tang ◽  
Wee Peng Tay ◽  
Edwin K P Chong
Keyword(s):  
Network Topology ◽  
Necessary Condition ◽  
Information Cascades ◽  
Real World Data ◽  
Distance Information ◽  
Statistical Knowledge ◽  
Current State ◽  
Reconstruction Methods ◽  
Vertex Set ◽  
General Graphs

Abstract We study the problem of inferring network topology from information cascades, in which the amount of time taken for information to diffuse across an edge in the network follows an unknown distribution. Unlike previous studies, which assume knowledge of these distributions, we only require that diffusion along different edges in the network be independent together with limited moment information (e.g. the means). We introduce the concept of a separating vertex set for a graph, which is a set of vertices in which for any two given distinct vertices of the graph there exists a vertex whose distance to them is different. We show that a necessary condition for reconstructing a tree perfectly using distance information between pairs of vertices is given by the size of an observed separating vertex set. We then propose an algorithm to recover the tree structure using infection times whose differences have means corresponding to the distance between two vertices. To improve the accuracy of our algorithm, we propose the concept of redundant vertices, which allows us to perform averaging to better estimate the distance between two vertices. Though the theory is developed mainly for tree networks, we demonstrate how the algorithm can be extended heuristically to general graphs. Simulations using synthetic and real networks and experiments using real-world data suggest that our proposed algorithm performs better than some current state-of-the-art network reconstruction methods.

scholarly journals Maximum Multiplicity of Matching Polynomial Roots and Minimum Path Cover in General Graphs

10.37236/525 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Minimum Size ◽  
Polynomial Roots ◽  
Path Cover ◽  
Minimum Path ◽  
Matching Polynomial ◽  
Minimum Number ◽  
Vertex Set ◽  
The Difference ◽  
General Graphs ◽  
Maximum Multiplicity

Let $G$ be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial $\mu(G,x)$ is at most the minimum number of vertex disjoint paths needed to cover the vertex set of $G$. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, we extend the notion of a $(\theta,G)$-extremal path cover (where $\theta$ is a root of $\mu(G,x)$) which was first introduced for trees to general graphs. Our proof makes use of the analogue of the Gallai-Edmonds Structure Theorem for general root. By way of contrast, we also show that the difference between the minimum size of a path cover and the maximum multiplicity of matching polynomial roots can be arbitrarily large.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

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