Korean J. Math. Vol. 23 No. 3 (2015) pp.401-408
DOI: https://doi.org/10.11568/kjm.2015.23.3.401

Maximum Zagreb indices in the class of $k$-apex trees

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Published: Sep 30, 2015
Keywords:
k-apex tree, First Zagreb index, Second Zagreb index.
Mathematics Subject Classification:
97K30, 94C15, 05C07.

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Tsend-Ayush Selenge
Department of Mathematics National University of Mongolia, Baga toiruu-1, Ulaanbaatar, Mongolia
Batmend Horoldagva
Department of Mathematics Mongolian National University of Education Baga toiruu-14, Ulaanbaatar, Mongolia

Abstract

The first and second Zagreb indices of a graph $G$ are defined as $M_1(G)=\sum_{v\in V} d_G(v)^2$ and $M_2(G)=\sum_{uv\in E(G)}d_G(u)\,d_G(v)\,,$ where $d_G(v)$ is the degree of the vertex $v$. $G$ is called a $k$-apex tree if $k$ is the smallest integer for which there exists a subset $X$ of $V(G)$ such that $|X|=k$ and $G-X$ is a tree. In this paper, we determine the maximum Zagreb indices in the class of all $k$-apex trees of order $n$ and characterize the corresponding extremal graphs.


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Supporting Agencies

This work is supported by the Korea Foundation for Advanced Studies’ Interna- tional Scholar Exchange Fellowship for the academic year of 2014–2015.

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