The Differential on Graph Operator Q(G)

If <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>,</mo> <mi>E</mi&...

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Main Authors: Ludwin A. Basilio, Jair Castro Simon, Jesús Leaños, Omar Rosario Cayetano
Format: Article
Language:English
Published: MDPI AG 2020-05-01
Series:Symmetry
Subjects:
Online Access:https://www.mdpi.com/2073-8994/12/5/751
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author Ludwin A. Basilio
Jair Castro Simon
Jesús Leaños
Omar Rosario Cayetano
author_facet Ludwin A. Basilio
Jair Castro Simon
Jesús Leaños
Omar Rosario Cayetano
author_sort Ludwin A. Basilio
collection DOAJ
description If <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>,</mo> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is a simple connected graph with the vertex set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and the edge set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, <i>S</i> is a subset of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of neighbors of <i>S</i> in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>∖</mo> <mi>S</mi> </mrow> </semantics> </math> </inline-formula>. Then, the differential of <i>S</i><inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mo>|</mo> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>|</mo> <mo>−</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> </mrow> </semantics> </math> </inline-formula>. The differential of <i>G</i>, denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, is the maximum value of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for all subsets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The graph operator <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as the graph that results by subdividing every edge of <i>G</i> once and joining pairs of these new vertices iff their corresponding edges are incident in <i>G</i>. In this paper, we study the relations between <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Besides, we exhibit some results relating the differential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.
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spelling doaj.art-a5e8e21f955c48e4a9a93e56163591602023-11-19T23:34:27ZengMDPI AGSymmetry2073-89942020-05-0112575110.3390/sym12050751The Differential on Graph Operator Q(G)Ludwin A. Basilio0Jair Castro Simon1Jesús Leaños2Omar Rosario Cayetano3Academic Unit of Mathematics, Autonomous University of Zacatecas, Paseo la Bufa, int. Calzada Solidaridad, Zacatecas 98060, MexicoFaculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita 39650, Acapulco, Guerrero, MexicoAcademic Unit of Mathematics, Autonomous University of Zacatecas, Paseo la Bufa, int. Calzada Solidaridad, Zacatecas 98060, MexicoFaculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita 39650, Acapulco, Guerrero, MexicoIf <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mo>=</mo> <mo>(</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>,</mo> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is a simple connected graph with the vertex set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and the edge set <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, <i>S</i> is a subset of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the set of neighbors of <i>S</i> in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>∖</mo> <mi>S</mi> </mrow> </semantics> </math> </inline-formula>. Then, the differential of <i>S</i><inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as <inline-formula> <math display="inline"> <semantics> <mrow> <mo>|</mo> <mi>B</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>|</mo> <mo>−</mo> <mo>|</mo> <mi>S</mi> <mo>|</mo> </mrow> </semantics> </math> </inline-formula>. The differential of <i>G</i>, denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, is the maximum value of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for all subsets <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The graph operator <inline-formula> <math display="inline"> <semantics> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is defined as the graph that results by subdividing every edge of <i>G</i> once and joining pairs of these new vertices iff their corresponding edges are incident in <i>G</i>. In this paper, we study the relations between <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mrow> <mo form="prefix">Q</mo> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Besides, we exhibit some results relating the differential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>∂</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.https://www.mdpi.com/2073-8994/12/5/751differential of a graphoperator graphsdifferential
spellingShingle Ludwin A. Basilio
Jair Castro Simon
Jesús Leaños
Omar Rosario Cayetano
The Differential on Graph Operator Q(G)
Symmetry
differential of a graph
operator graphs
differential
title The Differential on Graph Operator Q(G)
title_full The Differential on Graph Operator Q(G)
title_fullStr The Differential on Graph Operator Q(G)
title_full_unstemmed The Differential on Graph Operator Q(G)
title_short The Differential on Graph Operator Q(G)
title_sort differential on graph operator q g
topic differential of a graph
operator graphs
differential
url https://www.mdpi.com/2073-8994/12/5/751
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