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Sharp bounds for the Randić index of graphs with given minimum and maximum degree
The Randić index of a graph G, written R(G), is the sum of 1d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d
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Published in: | Discrete Applied Mathematics 2018-10, Vol.247, p.111-115 |
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container_title | Discrete Applied Mathematics |
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creator | O, Suil Shi, Yongtang |
description | The Randić index of a graph G, written R(G), is the sum of 1d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d |
doi_str_mv | 10.1016/j.dam.2018.03.064 |
format | article |
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Let d and D be positive integers d<D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G)≥dDd+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and maximum degree D, then R(G)≤n2−∑i=dD−1121i−1i+12; it is sharp for infinitely many n, and we characterize when equality holds in the bound.</description><identifier>ISSN: 0166-218X</identifier><identifier>EISSN: 1872-6771</identifier><identifier>DOI: 10.1016/j.dam.2018.03.064</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Branch & bound algorithms ; Combinatorics ; Generalized linear models ; Graph theory ; Integers ; Maximum degree ; Maximum likelihood method ; Minimum degree ; Randic index</subject><ispartof>Discrete Applied Mathematics, 2018-10, Vol.247, p.111-115</ispartof><rights>2018 Elsevier B.V.</rights><rights>Copyright Elsevier BV Oct 1, 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c325t-9e0786f11bfc1928277eeacf8fc6d4cbb383f3af62641f11f4fd533de2fb49cc3</citedby><cites>FETCH-LOGICAL-c325t-9e0786f11bfc1928277eeacf8fc6d4cbb383f3af62641f11f4fd533de2fb49cc3</cites><orcidid>0000-0001-9406-7967 ; 0000-0002-2182-6237</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>O, Suil</creatorcontrib><creatorcontrib>Shi, Yongtang</creatorcontrib><title>Sharp bounds for the Randić index of graphs with given minimum and maximum degree</title><title>Discrete Applied Mathematics</title><description>The Randić index of a graph G, written R(G), is the sum of 1d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d<D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G)≥dDd+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and maximum degree D, then R(G)≤n2−∑i=dD−1121i−1i+12; it is sharp for infinitely many n, and we characterize when equality holds in the bound.</description><subject>Branch & bound algorithms</subject><subject>Combinatorics</subject><subject>Generalized linear models</subject><subject>Graph theory</subject><subject>Integers</subject><subject>Maximum degree</subject><subject>Maximum likelihood method</subject><subject>Minimum degree</subject><subject>Randic index</subject><issn>0166-218X</issn><issn>1872-6771</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp9kMtKxDAUhoMoOI4-gLuA69ZcOmmLKxm8wYAwKrgLaXIyTbHtmLTj-AS-mA9mxnHtKifw_efyIXROSUoJFZdNalSbMkKLlPCUiOwATWiRs0TkOT1Ek8iIhNHi9RidhNAQQmj8TdDyqVZ-jat-7EzAtvd4qAEvVWfc9xd2nYEt7i1eebWuA_5wQ41XbgMdbl3n2rHFkcSt2v7WBlYe4BQdWfUW4OzvnaKX25vn-X2yeLx7mF8vEs3ZbEhKIHkhLKWV1bRkBctzAKVtYbUwma4qXnDLlRVMZDRiNrNmxrkBZqus1JpP0cW-79r37yOEQTb96Ls4UjJKs3heyUik6J7Svg_Bg5Vr71rlPyUlcqdONjKqkzt1knAZ1cXM1T4Dcf2NAy-DdtBpMM6DHqTp3T_pHxR5d6I</recordid><startdate>20181001</startdate><enddate>20181001</enddate><creator>O, Suil</creator><creator>Shi, Yongtang</creator><general>Elsevier B.V</general><general>Elsevier BV</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0001-9406-7967</orcidid><orcidid>https://orcid.org/0000-0002-2182-6237</orcidid></search><sort><creationdate>20181001</creationdate><title>Sharp bounds for the Randić index of graphs with given minimum and maximum degree</title><author>O, Suil ; Shi, Yongtang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c325t-9e0786f11bfc1928277eeacf8fc6d4cbb383f3af62641f11f4fd533de2fb49cc3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Branch & bound algorithms</topic><topic>Combinatorics</topic><topic>Generalized linear models</topic><topic>Graph theory</topic><topic>Integers</topic><topic>Maximum degree</topic><topic>Maximum likelihood method</topic><topic>Minimum degree</topic><topic>Randic index</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>O, Suil</creatorcontrib><creatorcontrib>Shi, Yongtang</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Discrete Applied Mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>O, Suil</au><au>Shi, Yongtang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Sharp bounds for the Randić index of graphs with given minimum and maximum degree</atitle><jtitle>Discrete Applied Mathematics</jtitle><date>2018-10-01</date><risdate>2018</risdate><volume>247</volume><spage>111</spage><epage>115</epage><pages>111-115</pages><issn>0166-218X</issn><eissn>1872-6771</eissn><abstract>The Randić index of a graph G, written R(G), is the sum of 1d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d<D. In this paper, we prove that if G is a graph with minimum degree d and maximum degree D, then R(G)≥dDd+Dn; equality holds only when G is an n-vertex (d,D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and maximum degree D, then R(G)≤n2−∑i=dD−1121i−1i+12; it is sharp for infinitely many n, and we characterize when equality holds in the bound.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.dam.2018.03.064</doi><tpages>5</tpages><orcidid>https://orcid.org/0000-0001-9406-7967</orcidid><orcidid>https://orcid.org/0000-0002-2182-6237</orcidid></addata></record> |
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subjects | Branch & bound algorithms Combinatorics Generalized linear models Graph theory Integers Maximum degree Maximum likelihood method Minimum degree Randic index |
title | Sharp bounds for the Randić index of graphs with given minimum and maximum degree |
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