{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T05:46:45Z","timestamp":1723096005380},"reference-count":0,"publisher":"Combinatorial Press","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2023,12,31]]},"abstract":"<jats:p>The harmonic index H ( G ) of a graph G is defined as the sum of the weights 2 d u + d v of all edges u v of G , where d u denotes the degree of a vertex u . Delorme et al. [1] (2002) put forward a conjecture concerning the minimum Randi\u0107 index among all connected graphs with n vertices and the minimum degree at least k . Motivated by this paper, a conjecture related to the minimum harmonic index among all connected graphs with n vertices and the minimum degree at least k was posed in [2]. In this work, we show that the conjecture is true for a connected graph on $n$ vertices with k vertices of degree n \u2212 1 , and it is also true for a k -tree. Moreover, we give a shorter proof of Liu\u2019s result [3].<\/jats:p>","DOI":"10.61091\/ars157-03","type":"journal-article","created":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T23:11:55Z","timestamp":1706051515000},"page":"39-44","source":"Crossref","is-referenced-by-count":0,"title":["On a conjecture of the harmonic index with given minimum degree of graphs and short proof of Liu\u2019s result on Randi\u0107 index"],"prefix":"10.61091","volume":"157","author":[{"name":"College of Mathematics and Statistics, Hunan Normal University, Changsha,Hunan 410081, P. R. China.","sequence":"first","affiliation":[]},{"given":"Hanyuan","family":"Deng","sequence":"first","affiliation":[]},{"given":"S.","family":"Balachandran","sequence":"additional","affiliation":[]},{"name":"Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa.","sequence":"additional","affiliation":[]},{"name":"Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur, India.","sequence":"additional","affiliation":[]},{"given":"S. Raja","family":"Balachandar","sequence":"additional","affiliation":[]},{"name":"Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur, India.","sequence":"additional","affiliation":[]}],"member":"39747","published-online":{"date-parts":[[2023,12,31]]},"container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T23:12:01Z","timestamp":1706051521000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-157\/on-a-conjecture-of-the-harmonic-index-with-given-minimum-degree-of-graphs-and-short-proof-of-lius-result-on-randic-index\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,31]]},"references-count":0,"URL":"https:\/\/doi.org\/10.61091\/ars157-03","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"type":"print","value":"0381-7032"},{"type":"electronic","value":"2817-5204"}],"subject":[],"published":{"date-parts":[[2023,12,31]]}}}