On Zagreb Energy of Certain Classes of Graphs (original) (raw)

New Results on Zagreb Energy of Graphs

Mathematical Problems in Engineering, 2021

Let G be a graph with vertex set V G = v 1 , … , v n , and let d i be the degree of v i . The Zagreb matrix of G is the square matrix of order n whose i , j -entry is equal to d i + d j if the vertices v i and v j are adjacent, and zero otherwise. The Zagreb energy ZE G of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we determine some classes of Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs.

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 α-adjacency energy of graphs and Zagreb index

AKCE Int. J. Graphs Comb., 2021

Let A(G)A(G)A(G) be the adjacency matrix and D(G)D(G)D(G) be the diagonal matrix of the vertex degrees of a simple connected graph GGG. Nikiforov defined the matrix Aalpha(G)A_{\alpha}(G)Aalpha(G) of the convex combinations of D(G)D(G)D(G) and A(G)A(G)A(G) as Aalpha(G)=alphaD(G)+(1−alpha)A(G)A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)Aalpha(G)=alphaD(G)+(1−alpha)A(G), for 0leqalphaleq10\leq \alpha\leq 10leqalphaleq1. If $ \rho_{1}\geq \rho_{2}\geq \dots \geq \rho_{n}$ are the eigenvalues of Aalpha(G)A_{\alpha}(G)Aalpha(G) (which we call alpha\alphaalpha-adjacency eigenvalues of GGG), the $ \alpha −adjacencyenergyof-adjacency energy of −adjacencyenergyofG$ is defined as EAalpha(G)=sumi=1nleft∣rhoi−frac2alphamnright∣E^{A_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i-\frac{2\alpha m}{n}\right|EAalpha(G)=sumi=1nleft∣rhoi−frac2alphamnright∣, where nnn is the order and mmm is the size of GGG. We obtain the upper and lower bounds for EAalpha(G)E^{A_{\alpha}}(G) EAalpha(G) in terms of order nnn, size mmm and Zagreb index Zg(G)Zg(G)Zg(G) associated to the structure of GGG. Further, we characterize the extremal graphs attaining these bounds.

On energy and Laplacian energy of bipartite graphs

Let G be a bipartite graph of order n with m edges. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for E(G) in terms of n, m, and det A. Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.

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.

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.

The hyper-Zagreb index 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...

Energy and Some Hamiltonian Properties of Graphs

2009

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [2]. For a graph G = (V, E), n := |V |, e := |E|, and G := (V, E), where E := { xy : x ∈ V, y ∈ V, x = y, xy ∈ E }. For a bipartite graph GBPT = (X, Y ; E), GBPT := (X, Y ; E ), where E := { xy : x ∈ X, y ∈ Y, xy ∈ E }. The degree of vertex vi is denoted by di. The concept of closure of a general graph G was introduced by Bondy and Chvátal [1]. The k closure of a graph G, denoted clk(G), is a graph obtained from G by recursively joining two nonadjacent vertices such that their degree sum is at least k. The idea for the closure of a balanced bipartite graph can be found in [1] and [6]. The k closure of a balanced bipartite graph GBPT = (X, Y ; E), where |X | = |Y |, denoted clk(GBPT ), is a graph obtained from G by recursively joining two nonadjacent vertices x ∈ X and y ∈ Y such that their degree sum is at least k. We use C(n, r) to denote the number of...

The Zagreb Indices and Some Hamiltonian Properties of Graphs

2020

Let G = (V, E) be a graph. The first Zagreb index and second Zagreb index of G are defined as v∈V d 2 (v) and uv∈E d(u)d(v), respectively. Using first and second Zagreb indices of graphs, we in this note present sufficient conditions for some Hamiltonian properties of graphs.