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Comparing Zagreb indices for connected graphs
Horoldagva, B.,Lee, S.G. North Holland ; Elsevier Science Ltd 2010 Discrete Applied Mathematics Vol.158 No.10
It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M<SUB>2</SUB>G)/m>=M<SUB>1</SUB>G)/n, where M<SUB>1</SUB>and M<SUB>2</SUB>are the first and second Zagreb indices. Hansen and Vukicevic proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m-n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M<SUB>2</SUB>(m-1)>M<SUB>1</SUB>n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.
Complete characterization of graphs for direct comparing Zagreb indices
Horoldagva, B.,Das, K.Ch.,Selenge, T.A. North Holland ; Elsevier Science Ltd 2016 Discrete Applied Mathematics Vol.215 No.-
<P>The classical first and second Zagreb indices of a graph G are defined as M-1(G) = Sigma(v is an element of V) d(G)(v)(2) and M-2(G) = Sigma(uv is an element of E(G)) d(G)(u) d(G)(V), where d(G)(v) is the degree of the vertex v of graph G. Recently, Furtula et al. (2014) studied the difference between the Zagreb indices and mentioned a problem to characterize the graphs for which M-1(G) > M-2 (G) or M-1(G) < M-2 (G) or M-1(G) = M-2 (G). In this paper we completely solve this problem. (C) 2016 Elsevier B.V. All rights reserved.</P>
MAXIMUM ZAGREB INDICES IN THE CLASS OF k-APEX TREES
SELENGE, TSEND-AYUSH,HOROLDAGVA, BATMEND The Kangwon-Kyungki Mathematical Society 2015 한국수학논문집 Vol.23 No.3
The first and second Zagreb indices of a graph G are defined as $M_1(G)={\sum}_{{\nu}{\in}V}d_G({\nu})^2$ and $M_2(G)={\sum}_{u{\nu}{\in}E(G)}d_G(u)d_G({\nu})$. where $d_G({\nu})$ is the degree of the vertex ${\nu}$. 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 ${\mid}X{\mid}$ = 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.