The edge signal number of a graph

2020 ◽  
pp. 2150024
Author(s):  
S. Balamurugan ◽  
R. Antony Doss
Keyword(s):  
Connected Graph ◽  
Minimum Order ◽  
Signal Set ◽  
Number Formula ◽  
Sum Formula ◽  
Distance Formula

For two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the signal distance [Formula: see text] from [Formula: see text] to [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is a path connecting [Formula: see text] and [Formula: see text], [Formula: see text] is the length of the path [Formula: see text] and in the sum [Formula: see text] runs over all the internal vertices between [Formula: see text] and [Formula: see text] in the path [Formula: see text]. A path between the vertices [Formula: see text] and [Formula: see text] of length [Formula: see text] is called a [Formula: see text] geosig path. A set [Formula: see text] is called a signal set, if every vertex [Formula: see text] in [Formula: see text] lies on a geosig path joining a pair of vertices of [Formula: see text]. The signal number [Formula: see text] is the minimum order of a signal set of a graph [Formula: see text]. An edge signal cover of [Formula: see text] is a set [Formula: see text] such that every edge of [Formula: see text] is contained in a geosig path joining some pair of vertices of [Formula: see text]. The edge signal number [Formula: see text] of [Formula: see text] is the minimum order of an edge signal cover and any edge signal cover of order [Formula: see text] is an edge signal basis of [Formula: see text]. In this paper, we initiate a study on the edge signal number of a graph [Formula: see text].

Upper triangle free detour number of a graph

2021 ◽  
pp. 2150094
Author(s):  
S. Sethu Ramalingam ◽  
S. Athisayanathan
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Proper Subset ◽  
Maximum Order ◽  
Minimum Order ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers ◽  
Distance Formula

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

The detour domination number of a graph

2017 ◽  
Vol 09 (01) ◽  
pp. 1750006 ◽  
Author(s):  
J. John ◽  
N. Arianayagam
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Order ◽  
Number Formula ◽  
Positive Integers ◽  
Domination Numbers

For a connected graph [Formula: see text], a set [Formula: see text] is called a detour dominating set of [Formula: see text], if [Formula: see text] is a detour set and dominating set of [Formula: see text]. The detour domination number [Formula: see text] of [Formula: see text] is the minimum order of its detour dominating sets and any detour dominating set of order [Formula: see text] is called a [Formula: see text] - set of [Formula: see text]. The detour domination numbers of some standard graphs are determined. Connected graph of order [Formula: see text] with detour domination number [Formula: see text] or [Formula: see text] is characterized. For positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph with [Formula: see text] and [Formula: see text].

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

On the minimum vertex covering transversal dominating sets in graphs and their classification

2017 ◽  
Vol 09 (05) ◽  
pp. 1750069 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Covering ◽  
Number Formula ◽  
Covering Set ◽  
Simple Connected Graph

Let [Formula: see text] be a simple and connected graph. A dominating set [Formula: see text] is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of [Formula: see text]. The vertex covering transversal domination number [Formula: see text] is the minimum cardinality among all vertex covering transversal dominating sets of [Formula: see text]. A vertex covering transversal dominating set of minimum cardinality [Formula: see text] is called a minimum vertex covering transversal dominating set or simply a [Formula: see text]-set. In this paper, we prove some general theorems on the vertex covering transversal domination number of a simple connected graph. We also provide some results about [Formula: see text]-sets and try to classify those sets based on their intersection with the minimum vertex covering sets.

scholarly journals The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums

2021 ◽  
pp. 1-18
Author(s):  
Yana Niu ◽  
Rong Ma ◽  
Yulong Zhang ◽  
Peilin Jiang
Keyword(s):  
Asymptotic Formula ◽  
Real Number ◽  
Analytic Continuation ◽  
Dirichlet Character ◽  
Mean Value ◽  
Gauss Sums ◽  
Number Formula ◽  
Sum Formula ◽  
The Mean ◽  
Generalized Dirichlet

Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

The open detour number of a graph

2020 ◽  
pp. 2050088
Author(s):  
J. John ◽  
V. R. Sunil Kumar
Keyword(s):  
Connected Graph ◽  
Internal Vertex ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Geodetic Number ◽  
Number Formula ◽  
Positive Integers

A set [Formula: see text] is called an open detour set of [Formula: see text] if for each vertex [Formula: see text] in [Formula: see text], either (1) [Formula: see text] is a detour simplicial vertex of [Formula: see text] and [Formula: see text] or (2) [Formula: see text] is an internal vertex of an [Formula: see text]-[Formula: see text] detour for some [Formula: see text]. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of [Formula: see text], denoted by [Formula: see text]. Connected graphs of order [Formula: see text] with open detour number [Formula: see text] or [Formula: see text] are characterized. It is shown that for any two positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the detour number of [Formula: see text]. It is also shown that for every pair of positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the open geodetic number of [Formula: see text].

A Nordhaus–Gaddum bound for Roman domination

2019 ◽  
Vol 11 (05) ◽  
pp. 1950055
Author(s):  
Nader Jafari Rad ◽  
Hadi Rahbani
Keyword(s):  
Connected Graph ◽  
Domination Number ◽  
Discrete Math ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Graph ◽  
Number Formula ◽  
Roman Dominating Function ◽  
Dominating Function

A Roman dominating function of a graph [Formula: see text] is a labeling [Formula: see text] such that every vertex with label [Formula: see text] has a neighbor with label [Formula: see text]. The Roman domination number, [Formula: see text] of [Formula: see text], is the minimum of [Formula: see text] over such functions. Let [Formula: see text] be an [Formula: see text]-vertex graph. Chambers et al. [E. W. Chambers, B. Kinnersley, N. Prince and D. B. West, External Problems for Roman domination Siam J. Discrete Math. 23 (2009) 1575–1586.] proved that if [Formula: see text] is a connected graph of order [Formula: see text], then [Formula: see text], with equality if and only if [Formula: see text] or [Formula: see text] is [Formula: see text] or [Formula: see text]. In this paper, we construct a specific family of graphs [Formula: see text], and prove that if [Formula: see text] and [Formula: see text], then [Formula: see text], and this bound is sharp.

Herscovici’s conjecture on C2n × G

2020 ◽  
Vol 12 (05) ◽  
pp. 2050071
Author(s):  
A. Lourdusamy ◽  
T. Mathivanan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Connected Graphs ◽  
Number Formula ◽  
Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

Italian domination on Mycielskian and Sierpinski graphs

2020 ◽  
pp. 2150037
Author(s):  
Jismy Varghese ◽  
S. Aparna Lakshmanan
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Sierpiński Graphs ◽  
Number Formula ◽  
Sum Formula ◽  
The Impact ◽  
Edge Addition ◽  
Dominating Function

An Italian dominating function (IDF) of a graph G is a function [Formula: see text] satisfying the condition that for every [Formula: see text] with [Formula: see text] The weight of an IDF on [Formula: see text] is the sum [Formula: see text] and Italian domination number, [Formula: see text] is the minimum weight of an IDF. In this paper, we prove that [Formula: see text] where [Formula: see text] is the Mycielskian graph of [Formula: see text]. We have also studied the impact of edge addition on Italian domination number. We also obtain a bound for the Italian domination number of Sierpinski graph [Formula: see text] and find the exact value of [Formula: see text].

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