β-Differential of a Graph

Basilio, Ludwin A.; Bermudo, Sergio; Leaños, Jesús; Sigarreta, José M.

doi:10.3390/sym9100205

Cited by 8 publications

(12 citation statements)

References 17 publications

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“…The authors in [1,2,[4][5][6][7][8]10] gave a lot of bounds on ∂(G), but they did not obtain many on the cardinality of the differential sets. Now, considering the possible applications mentioned in the introduction, it seems convenient to find them.…”

Section: Some Bounds For the Cardinality Of Smentioning

confidence: 99%

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General Properties on Differential Sets of a Graph

Basilio¹,

Bermudo²,

Hernández-Gómez³

et al. 2021

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Let G=(V,E) be a graph, and let β∈R. Motivated by a service coverage maximization problem with limited resources, we study the β-differential of G. The β-differential of G, denoted by ∂β(G), is defined as ∂β(G):=max{|B(S)|−β|S|suchthatS⊆V}. The case in which β=1 is known as the differential of G, and hence ∂β(G) can be considered as a generalization of the differential ∂(G) of G. In this paper, upper and lower bounds for ∂β(G) are given in terms of its order |G|, minimum degree δ(G), maximum degree Δ(G), among other invariants of G. Likewise, the β-differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have.

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Section: Some Bounds For the Cardinality Of Smentioning

confidence: 99%

“…The differential of a graph and other kinds of differentials of a set were introduced in [2] , where dimensions are given for the differential of a graph. Additionally, this parameter has been studied in [1,[3][4][5][6][7][8][9]. In addition, the differential of the Cartesian product of graphs was studied in [10].…”

Section: Introductionmentioning

confidence: 99%

General Properties on Differential Sets of a Graph

Basilio¹,

Bermudo²,

Hernández-Gómez³

et al. 2021

Axioms

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“…The differential of a set could act as a measure of how this set can influence the rest of the elements ( [4]). In [3], the β-differential of a graph G is defined as ( ) : max{ ( ) :…”

Section: Introductionmentioning

confidence: 99%

The β-differential for the Square Lattice

2021

Proceedings of 2021 the 11th International Workshop on Computer Science and Engineering

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“…The differential ∂(G) of a graph was also analyzed in [3,[5][6][7][8][9][10][11][12][13]. In [10], it was established that…”

Section: Introductionmentioning

confidence: 99%

“…Most of the contributions reported in [3,[5][6][7][8][9][10][11][12][13] lied on at least one of the following issues of the differential of a graph: I.…”

Section: Introductionmentioning

confidence: 99%

The Differential on Graph Operator Q(G)

et al. 2020

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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.

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β-Differential of a Graph

Abstract: Let G = (V, E) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B(D) be the set of neighbours ofMotivated by an influential maximization problem, in this paper we study the β-differential of G.

Cited by 8 publications

References 17 publications

General Properties on Differential Sets of a Graph

General Properties on Differential Sets of a Graph

The β-differential for the Square Lattice

The Differential on Graph Operator Q(G)

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