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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Format: | Article |
Language: | English |
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MDPI AG
2020-05-01
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Series: | Symmetry |
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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. |
first_indexed | 2024-03-10T20:01:41Z |
format | Article |
id | doaj.art-a5e8e21f955c48e4a9a93e5616359160 |
institution | Directory Open Access Journal |
issn | 2073-8994 |
language | English |
last_indexed | 2024-03-10T20:01:41Z |
publishDate | 2020-05-01 |
publisher | MDPI AG |
record_format | Article |
series | Symmetry |
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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