The Zagreb Indices and Some Hamiltonian Properties of Graphs

2019

In this paper, only finite undirected graphs without loops or multiple edges are considered. Notation and terminology not defined here follow that described in [2]. Let G = (V (G), E(G)) be a graph. Denote by n, m, δ, and κ the order, size, minimum degree, and connectivity of G, respectively. The complement of G is denoted by G. The hyper-Zagreb index of G, denoted HZ(G), is defined as ∑ uv∈E(G)(dG(u) + dG(v)) 2 (see [11]). It needs to be mentioned here that the hyper-Zagreb index of G is actually equal to F (G) + 2M2(G), where F (G) is the forgotten topological index of G (see [5]) and M2(G) is the second Zagreb index of G (see [9]). Denote by μn(G) the largest eigenvalue of the adjacency matrix of a graph G of order n. For two disjoint graphs G1 and G2, the union and join of G1 and G2 are denoted by G1 + G2 and G1 ∨ G2, respectively. Denote by sK1 the union of s isolated vertices. The concept of closure of a graph G was introduced by Bondy and Chvátal in [1]. The k-closure of a gr...

Hacettepe Journal of Mathematics and Statistics, 2019

Recently, Furtula et al. [B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 2014] introduced a new vertex-degree-based graph invariant "reduced second Zagreb index" in chemical graph theory. Here we generalize the reduced second Zagreb index (call "general reduced second Zagreb index"), denoted by GRM α (G) and is defined as: , where α is any real number and d G (v) is the degree of the vertex v of G. Let G k n be the set of connected graphs of order n with k cut edges. In this paper, we study some properties of GRM α (G) for connected graphs G. Moreover, we obtain the sharp upper bounds on GRM α (G) in G k n for α ≥ -1/2 and characterize the extremal 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.

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

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.

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.

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.

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.

   , 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.

AKCE International Journal of Graphs and Combinatorics, 2015

For a nontrivial connected graph G, its Harary index H (G) is defined as  {u, v}⊆V (G) 1 d G (u, v) , where d G (u, v) is the distance between vertices u and v. Hua and Wang (2013), using Harary index, obtained a sufficient condition for the traceable graphs. In this note, we use Harary index to present sufficient conditions for Hamiltonian and Hamilton-connected graphs. c