scholarly journals The Differential on Graph Operator Q(G)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 751
Author(s):  
Ludwin Basilio ◽  
Jair Simon ◽  
Jesús Leaños ◽  
Omar Cayetano
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Vertex Cover ◽  
Domination Number ◽  
Graph Invariants ◽  
Maximum Value ◽  
Vertex Set ◽  
Simple Connected Graph

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

scholarly journals New bounds on the rainbow domination subdivision number

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 615-622 ◽  
Author(s):  
Mohyedin Falahat ◽  
Seyed Sheikholeslami ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Domination Subdivision Number ◽  
Dominating Function

A 2-rainbow dominating function (2RDF) of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v ? V(G) with f (v) = ? the condition Uu?N(v) f(u)= {1,2} is fulfilled, where N(v) is the open neighborhood of v. The weight of a 2RDF f is the value ?(f) = ?v?V |f(v)|. The 2-rainbow domination number of a graph G, denoted by r2(G), is the minimum weight of a 2RDF of G. The 2-rainbow domination subdivision number sd?r2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the 2-rainbow domination number. In this paper we prove that for every simple connected graph G of order n ? 3, sd?r2(G)? 3 + min{d2(v)|v?V and d(v)?2} where d2(v) is the number of vertices of G at distance 2 from v.

On the sum of the generalized distance eigenvalues of graphs

2020 ◽  
pp. 2050100
Author(s):  
Hilal A. Ganie ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Distance Matrix ◽  
Wiener Index ◽  
Index Formula ◽  
Graph Invariants ◽  
Matrix Formula ◽  
Simple Connected Graph ◽  
Eigenvalues Of Graphs ◽  
Generalized Distance

In this paper, we study the generalized distance matrix [Formula: see text] assigned to simple connected graph [Formula: see text], which is the convex combinations of Tr[Formula: see text] and [Formula: see text] and defined as [Formula: see text] where [Formula: see text] and Tr[Formula: see text] denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph [Formula: see text], respectively. Denote with [Formula: see text], the generalized distance eigenvalues of [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text] be, respectively, the sum of [Formula: see text]-largest generalized distance eigenvalues and the sum of [Formula: see text]-smallest generalized distance eigenvalues of [Formula: see text]. We obtain bounds for [Formula: see text] and [Formula: see text] in terms of the order [Formula: see text], the Wiener index [Formula: see text] and parameter [Formula: see text]. For a graph [Formula: see text] of diameter 2, we establish a relationship between the [Formula: see text] and the sum of [Formula: see text]-largest generalized adjacency eigenvalues of the complement [Formula: see text]. We characterize the connected bipartite graph and the connected graphs with given independence number that attains the minimum value for [Formula: see text]. We also obtain some bounds for the graph invariants [Formula: see text] and [Formula: see text].

THE SIGNED STAR DOMINATION NUMBER OF CAYLEY GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250017 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
G. KALAIMURUGAN ◽  
WELL Y. CHOU
Keyword(s):  
Connected Graph ◽  
Cayley Graphs ◽  
Domination Number ◽  
Vertex Set ◽  
Simple Connected Graph ◽  
Dominating Function

Let G be a simple connected graph with vertex set V(G) and edge set E(G). A function f : E(G) → {-1, 1} is called a signed star dominating function (SSDF) on G if ∑e∈E(v) f(e) ≥ 1 for every v ∈ V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as γ SS (G) = min {∑e∈E(G) f(e) | f is a SSDF on G}. In this paper, we obtain exact values for the signed star domination number for certain classes of Cayley digraphs and Cayley graphs.

Edge (m,k)-choosability of graphs

2021 ◽  
Author(s):  
P. Soorya ◽  
K. A. Germina
Keyword(s):  
Connected Graph ◽  
Matching Number ◽  
Graph Theoretic ◽  
Maximum Value ◽  
Choice Number ◽  
The Given ◽  
Simple Connected Graph

Let [Formula: see text] be a simple, connected graph of order [Formula: see text] and size [Formula: see text] Then, [Formula: see text] is said to be edge [Formula: see text]-choosable, if there exists a collection of subsets of the edge set, [Formula: see text] of cardinality [Formula: see text] such that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are incident. This paper initiates a study on edge [Formula: see text]-choosability of certain fundamental classes of graphs and determines the maximum value of [Formula: see text] for which the given graph [Formula: see text] is edge [Formula: see text]-choosable. Also, in this paper, the relation between edge choice number and other graph theoretic parameters is discussed and we have given a conjecture on the relation between edge choice number and matching number of a graph.

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.

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].

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 Nonessential sum graph of an Artinian ring

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bikash Barman ◽  
Kukil Kalpa Rajkhowa
Keyword(s):  
Design Methodology ◽  
Undirected Graph ◽  
Independence Number ◽  
Domination Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Content Type ◽  
Cut Vertex ◽  
Vertex Set ◽  
Regular Character

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.

scholarly journals Co-Secure Set Domination in Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 66-68
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Secure Set ◽  
Domination In Graphs

Throughout this paper, consider G = (V,E) as a connected graph. A subset D of V(G) is a set dominating set of G if for every M  V / D there exists a non-empty set N of D such that the induced sub graph <MUN> is connected. A subset D of the vertex set of a graph G is called a co-secure dominating set of a graph if D is a dominating set, and for each u' D there exists a vertex v'V / D such that u'v' is an edge and D \u'v' is a dominating set. A co-secure dominating set D is a co-secure set dominating set of G if D is also a set dominating set of G. The co-secure set domination number G s cs γ is the minimum cardinality of a co-secure set dominating set. In this paper we initiate the study of this new parameter & also determine the co-secure set domination number of some standard graphs and obtain its bounds.

On the rainbow domination subdivision numbers in graphs

2016 ◽  
Vol 09 (01) ◽  
pp. 1650018 ◽  
Author(s):  
N. Dehgardi ◽  
M. Falahat ◽  
S. M. Sheikholeslami ◽  
Abdollah Khodkar
Keyword(s):  
Connected Graph ◽  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Domination Subdivision Number ◽  
Dominating Function

A [Formula: see text]-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of G. The [Formula: see text]-rainbow domination subdivision number [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the 2-rainbow domination number. It is conjectured that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. In this paper, we first prove this conjecture for some classes of graphs and then we prove that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text].

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