On the Zagreb index inequality of graphs with prescribed vertex degrees (original) (raw)
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The Bounds for the First General Zagreb Index of a Graph
Universal Journal of Mathematics and Applications, 2021
The first general Zagreb index of a graph G is defined as the sum of the αth powers of the vertex degrees of G, where α is a real number such that α = 0 and α = 1. In this note, for α > 1, we present upper bounds involving chromatic and clique numbers for the first general Zagreb index of a graph; for an integer α ≥ 2, we present a lower bound involving the independence number for the first general Zagreb index of a graph.
Some Inequalities for the First General Zagreb Index of Graphs and Line Graphs
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.
Upper bounds for the reduced second zagreb index of graphs
2021
The graph invariant RM2RM_2RM2, known under the name reduced second Zagreb index, is defined as RM2(G)=sumuvinE(G)(dG(u)−1)(dG(v)−1)RM_2(G)=sum_{uvin E(G)}(d_G(u)-1)(d_G(v)-1)RM2(G)=sumuvinE(G)(dG(u)−1)(dG(v)−1), where dG(v)d_G(v)dG(v) is the degree of the vertex vvv of the graph GGG. In this paper, we give a tight upper bound of RM_2RM_2RM2 for the class of graphs of order nnn and size mmm with at least one dominating vertex. Also, we obtain sharp upper bounds on RM2RM_2RM2 for all graphs of order nnn with kkk dominating vertices and for all graphs of order nnn with kkk pendant vertices. Finally, we give a sharp upper bound on RM2RM_2RM_2 for all kkk-apex trees of order nnn. Moreover, the corresponding extremal graphs are characterized.
On Comparing Zagreb Indices 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.
On general reduced second Zagreb index of graphs
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: GRM α (G) = uv∈E(G) (d G (u) + α)(d G (v) + α), 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.
2014
In the mathematical discipline of Graph Theory, labeling of graphs is a mapping that sends the edges or vertices or both of a graph to the set of numbers or subset of a set under certain conditions. Ever since the introduction of the concept labeling of graphs, it has been an active area of research in Graph Theory. Most graph labelings trace their origin to a paper by Rosa [14]. In 1983, B. D. Acharya [1] initiated a general study of the labeling of the vertices and edges of a graph using the subset of a set known as set-valuation of graphs and indicated their potential applications in different areas of human inquiry. He introduced the concept of set-indexer of a graph and proved that every graph admits a set-indexer. Acharya also pioneered the notion of set-indexing number of a graph developing the classes of set-graceful as well as set-semigraceful graphs. Meanwhile, Acharya and Hegde [5] undertook the study of yet another notion of set-valuation of graphs called set-sequential ...
Comparing Zagreb indices for connected graphs
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.
Maximizing the Zagreb indices of (n;m)-graphs
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.
Some Upper Bounds on the First General Zagreb Index
Journal of Mathematics
The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ < 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.