The first reformulated Zagreb index EM1(G) of a simple graph G is defined as the sum of the terms (du+dv-2)2 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.
{"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}
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.
具有|V|顶点和|E|边的图G上的局部反幻标记被定义为赋值f: E ->{1,2，…， |E|}使得任意两个相邻的顶点u和v的权值不同，即w(u) = w(v)，其中w(u) = Σe∈E(u) f(E)， E(u)是与u相关的边的集合。因此，任何局部反奇异标记都可以导出G的适当顶点着色，其中顶点u被赋予w(u)的颜色。局部反幻色数用χla(G)表示，它是由G的局部反幻标记所引起的所有色所占的最小色数。本文给出了单环图的局部反幻色数，即只包含一个环的图，如有两个相邻的环的风筝图和环图。
{"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}
Let Tn be the set of all trees with n ≤ 10 vertices. We show that the Laplacian energy of any tree Tn is strictly between the Laplacian energy of the path Pn and the star Sn, partially proving the conjecture that this hold for any tree.
{"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}
For a colour cluster C = (C1, C2, C3, …, Cℓ), where Ci is a colour class such that ∣Ci∣ = ri, a positive integer, we investigate two types of simple connected graph structures G1C, G2C which represent graphical embodiments of the colour cluster such that the chromatic numbers χ(G1C) = χ(G2C) = ℓ and $min{varepsilon(G^{C}_1)}=min{varepsilon(G^{C}_2)} =sumlimits_{i=1}^{ell}r_i-1$, and ɛ(G) is the size of a graph G. In this paper, we also discuss the chromatic Zagreb indices corresponding to G1C, G2C.
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：