On general reduced second Zagreb index of graphs

Hacettepe Journal of Mathematics and Statistics, 2012

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. It is well-known that for connected or disconnected graphs, M2/m ≥ M1/n does not hold always. In K. C. Das (On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010), it has been shown that the above relation holds for a special kind of graph. Here we continue our search for special kinds of graph for which the above relation holds.

2021

The graph invariant $RM_2$‎, ‎known under the name reduced second Zagreb index‎, ‎is defined as $RM_2(G)=sum_{uvin E(G)}(d_G(u)-1)(d_G(v)-1)$‎, ‎where $d_G(v)$ is the degree of the vertex $v$ of the graph $G$‎. ‎In this paper‎, ‎we give a tight upper bound of $RM_2$ for the class of graphs of order $n$ and size $m$ with at least one dominating vertex‎. ‎Also‎, ‎we obtain sharp upper bounds on $RM_2$ for all graphs of order $n$ with $k$ dominating vertices and for all graphs of order $n$ with $k$ pendant vertices‎. ‎Finally‎, ‎we give a sharp upper bound on $RM_2$ for all $k$-apex trees of order $n$‎. ‎Moreover‎, ‎the corresponding extremal graphs are characterized‎.

Discrete Applied Mathematics, 2010

It was conjectured that for each simple graph G = (V , E) with n = |V (G)| vertices and m = |E(G)| edges, it holds M 2 (G)/m ≥ M 1 (G)/n, where M 1 and M 2 are the first and second Zagreb indices. Hansen and Vukičević 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 2 /(m − 1) > M 1 /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.

MATCH Communications in Mathematical and in Computer Chemistry

For a (molecular) graph, the first and second Zagreb indices (M1 and M2) are two well-known topological indices in chemical graph theory introduced in 1972 by Gutman and Trinajstić. Let Gn,m be the set of connected graphs of order n and with m edges. In this paper we characterize the extremal graphs from Gn,m with n + 2 ≥ m ≥ 2n-4 with maximal first Zagreb index and from Gn,m with m-n = (k2)-k for k ≥ 4 with maximal second Zagreb index, respectively. Finally a related conjecture has been proposed to the extremal graphs with respect to second Zagreb index.

Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2020

The first general Zagreb index M a 1 ðGÞ of a graph G is equal to the sum of the ath powers of the vertex degrees of G. For a ! 0 and k ! 1, we obtain the lower and upper bounds for M a 1 ðGÞ and M a 1 ðLðGÞÞ in terms of order, size, minimum/maximum vertex degrees and minimal nonpendant vertex degree using some classical inequalities and majorization technique, where L(G) is the line graph of G. Also, we obtain some bounds and exact values of M a 1 ðJðGÞÞ and M a 1 ðL k ðGÞÞ, where J(G) is a jump graph (complement of a line graph) and L k ðGÞ is an iterated line graph of a graph G.

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2016

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 d G (v) is the degree of the vertex v. In this paper we compute these indices for link and splice of graphs. In continuation, with use these graph operations, we compute the first and the second multiplicative Zagreb indices for a class of dendrimers.

In this paper, we introduce Zagreb Indices of Some New Graphs. Exactly, first index, second index and forgotten index. New graphs are generated from the initial graphs by graph operations. We also created some possible applications on the Zagreb indices as special cases.

2012

In this study, we first find formulae for the first and second Zagreb indices and coindices of certain classical graph types including path, cycle, star and complete graphs. Secondly we give similar formulae for the first and second Zagreb coindices.

Iraqi journal of science, 2020

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds. Let be a simple connected graph. The Hyper-Zagreb index of the graph , is defined as ,where and are the degrees of vertex and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .