{"title":"Four new operations related to composition and their reformulated Zagreb index","authors":"K. Pattabiraman, A. Santhakumar","doi":"10.19184/IJC.2018.2.1.5","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.1.5","url":null,"abstract":"The first reformulated Zagreb index EM<sub>1</sub>(G) of a simple graph G is defined as the sum of the terms (d<sub>u</sub>+d<sub>v</sub><span style=\"font-family: symbol;\">-</span>2)<sup>2</sup> over all edges uv of G. In 2017, Sarala et al. introduced four new operations (F-product) of graphs. In this paper, we study the first reformulated Zagreb index for the F-product of some special well-known graphs such as subdivision and total graph.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"100 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79330288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local antimagic vertex coloring of unicyclic graphs","authors":"N. H. Nazula, S. Slamin, D. Dafik","doi":"10.19184/ijc.2018.2.1.4","DOIUrl":"https://doi.org/10.19184/ijc.2018.2.1.4","url":null,"abstract":"The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈E(u) f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75901326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Laplacian energy of trees with at most 10 vertices","authors":"M. U. Rehman, M. Ajmal, T. Kamran","doi":"10.19184/IJC.2018.2.1.3","DOIUrl":"https://doi.org/10.19184/IJC.2018.2.1.3","url":null,"abstract":"Let T<sub>n</sub> be the set of all trees with n ≤ 10 vertices. We show that the Laplacian energy of any tree T<sub>n</sub> is strictly between the Laplacian energy of the path Pn and the star S<sub>n</sub>, partially proving the conjecture that this hold for any tree.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79110936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Chromatic Zagreb indices for graphical embodiment of colour clusters","authors":"J. Kok, S. Naduvath, M. Jamil","doi":"10.19184/IJC.2019.3.1.6","DOIUrl":"https://doi.org/10.19184/IJC.2019.3.1.6","url":null,"abstract":"<p>For a colour cluster <span class=\"math\"><em>C</em> = (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, …, C<sub>ℓ</sub>)</span>, where <span class=\"math\">C<sub><em>i</em></sub></span> is a colour class such that <span class=\"math\">∣C<sub><em>i</em></sub>∣ = <em>r</em><sub><em>i</em></sub></span>, a positive integer, we investigate two types of simple connected graph structures <span class=\"math\"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class=\"math\"><em>G</em><sub>2</sub><sup><em>C</em></sup></span> which represent graphical embodiments of the colour cluster such that the chromatic numbers <span class=\"math\"><em>χ</em>(<em>G</em><sub>1</sub><sup><em>C</em></sup>) = <em>χ</em>(<em>G</em><sub>2</sub><sup><em>C</em></sup>) = ℓ</span> and <span class=\"math\">$min{varepsilon(G^{C}_1)}=min{varepsilon(G^{C}_2)} =sumlimits_{i=1}^{ell}r_i-1$</span>, and <span class=\"math\"><em>ɛ</em>(<em>G</em>)</span> is the size of a graph <span class=\"math\"><em>G</em></span>. In this paper, we also discuss the chromatic Zagreb indices corresponding to <span class=\"math\"><em>G</em><sub>1</sub><sup><em>C</em></sup></span>, <span class=\"math\"><em>G</em><sub>2</sub><sup><em>C</em></sup></span>.</p>","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"112 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80656023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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