<abstract xmlns="http://www.w3.org/1999/xhtml"> For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2016-0008_001.jpg"></inline-graphic>, where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2016-0008_002.jpg"></inline-graphic> and <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2016-0008_003.jpg"></inline-graphic> named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2016-0008_004.jpg"></inline-graphic>. The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs. </abstract>

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants  and  named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"On the first Zagreb index and multiplicative Zagreb coindices of graphs"}

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G....

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Publié en ligne: 21 sept. 2017

Pages: 153 - 176

Reçu: 05 mars 2014

Accepté: 10 févr. 2014

DOI: https://doi.org/10.1515/auom-2016-0008

Mots clés
First Zagreb index, First and Second multiplicative Zagreb coindex, Narumi-Katayama index

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Publié en ligne: 21 sept. 2017

Pages: 153 - 176

Reçu: 05 mars 2014

Accepté: 10 févr. 2014

DOI: https://doi.org/10.1515/auom-2016-0008

Mots clésFirst Zagreb index, First and Second multiplicative Zagreb coindex, Narumi-Katayama index

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
First Zagreb index, First and Second multiplicative Zagreb coindex, Narumi-Katayama index