Batmend Horoldagva

• PhD
• Professor at MNUE, Ulaanbaatar, Mongolia

The generalized ABC index of a graph G, denoted by ABC α , is defined as the sum of the terms [(d(v) + d(u) − 2)/d(v)d(u)] α over all pairs of adjacent vertices, where d(u) is the degree of the vertex u and α is a real number. In this paper, we prove that for α ≤ −1, the balanced double broom is the unique tree that minimizes ABC α among trees of o...

Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph $G$. The three set of graphs are as fo...

Recently, Gutman defined a new vertex-degree-based graph invariant, named the Sombor index $SO$ of a graph $G$, and is defined by $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2},$$ where $d_G(v)$ is the degree of the vertex $v$ of $G$. In this paper, we obtain the sharp lower and upper bounds on $SO(G)$ of a connected graph, and characterize grap...

The exponential second Zagreb index of a graph G is defined as eM2(G)=∑xy∈E(G)edxdy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{M_2}(G)=\sum _{xy\in E(G)}e^{d_xd_...

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRMα, known under the name general reduced second Zagreb index, is defined...

The irregularity of a graph is the sum of the absolute values of the differences of degrees of pairs of adjacent vertices. In this paper, we obtain an upper bound on the irregularity of graphs in terms of the order, the size and the number of pendant vertices, and characterize the extremal graphs. Moreover, we characterize the graphs with maximum i...

The concept of Sombor indices (SO) of a graph was recently introduced by Gutman and the reduced Sombor index [Formula: see text] of a graph [Formula: see text] is defined by [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text]. In this paper, we study the extremal properties of the reduced Sombor index and c...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity avec(G) of a graph G is the mean value of eccentricities of all vertices of G. In this paper, we introduce some transformations for connected graphs that do not decrease their average eccentricity. Using these transformations, we completely d...

The graph invariant RM2, known under the name reduced second Zagreb index, is defined as (Formula presented), where dG(v) is the degree of the vertex v of the graph G. In this paper, we give a tight upper bound of RM2 for the class of graphs of order n and size m with at least one dominating vertex. Also, we obtain sharp upper bounds on RM2 for all...

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in G. In this paper, we introduce several graph transformations that are useful tools for the study of the extremal properties of the multiplic...

Recently, Gutman defined a new vertex-degree-based graph invariant, named the Sombor index $SO$ of a graph $G$, and is defined by $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2},$$ where $d_G(v)$ is the degree of the vertex $v$ of $G$. In this paper, we obtain the sharp lower and upper bounds on $SO(G)$ of a connected graph, and characterize gra...

Recently, Gutman introduced the class of stepwise irregular graphs and studied their properties. A graph is stepwise irregular if the difference between the degrees of any two adjacent vertices is exactly one. In this paper, we get some upper bounds on the maximum degree and sharp upper bounds on the size of stepwise irregular graphs. Furthermore,...

Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph $G$. The three set of graphs are as fo...

The multiplicative sum Zagreb index of a graph G, denoted by Π * 1 (G), is the product of the sum of the degrees of adjacent vertices in G. This graphical invariant is the multiplicative version of the well known first Zagreb index and introduced by Eliasi, Iranmanesh and Gutman (MATCH Commun. Math. Comput. Chem. 68 (2012) 217-230). In this paper w...

Given a graph G = (V, E), the variable first and second Zagreb indices are defined by λ M 1 (G) = vi∈V d 2λ i and λ M 2 (G) = vivj ∈E d λ i · d λ j , where d i is the degree of the vertex v i and λ is any real number. Let G ν be the class of connected graphs with cyclomatic number ν (ν ≥ 1). In this paper, we give a lower bound on λ M 2 (G) − λ M 1...

Let G be a simple connected graph of order n ≥ 9. Let q2 be the second largest signless Laplacian eigenvalue of G and λ1 be the index of G. Cvetković et al. [Publ. Inst. Math. (Beograd) 81(95) (2007) 11-27] conjectured that 1 − √ n − 1 ≤ q2 − λ1 ≤ n − 2 − √ 2n − 4, where the left equality holds if and only if G is the star K1,n−1, and the right equ...

Let G be a simple connected graph of order n ≥ 9. Let q2 be the second largest signless Laplacian eigenvalue of G and λ1 be the index of G. Cvetković et al. [Publ. Inst. Math. (Beograd) 81(95) (2007) 11-27] conjectured that 1 − √ n − 1 ≤ q2 − λ1 ≤ n − 2 − √ 2n − 4, where the left equality holds if and only if G is the star K1,n−1, and the right equ...

The reduced second Zagreb index \(RM_2\) of a graph G is defined as \(RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)\), where \(d_G(u)\) is the degree of the vertex u of graph G. Furtula et al. (Discrete Appl Math 178: 83–88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to \(RM_2\...

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(G) and is...

Let G=(V,E) be a graph with vertex set V and edge set E. The ve-degree of a vertex v∈V equals the number of edges ve-dominated by v and the ev-degree of an edge e∈E equals the number of vertices ev-dominated by e. Recently, Chellali et al. studied the properties of ve-degree and ev-degree of graphs (Chellali et al., 2017). Also they focused on the...

The classical first and second Zagreb indices of a graph G are defined as M1(G) = Σv∈V dG(v)² and M2(G) = Σuv∈(G) dG(u) dG(v), where dG(v) is the degree of the vertex v of graph G. The reduced second Zagreb index of a graph G is defined as MR2(G) = Σuv∈E(G) (dG(u)-1)(dG(v)-1). Recently, the reduced second Zagreb index and difference of Zagreb indic...

The classical first and second Zagreb indices of a graph are defined as and , where is the degree of the vertex of graph . Recently, Furtula et al. (2014) studied the difference between the Zagreb indices and mentioned a problem to characterize the graphs for which or or . In this paper we completely solve this problem.

The irregularity index of a simple graph G is the number of distinct elements in the degree sequence of G. If the irregularity index of a connected graph G is equal to the maximum vertex degree, then G is said to be maximally irregular. In this paper, we determine the maximum size of maximally irregular graphs with given order and irregularity inde...

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 tr...

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 of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cyc...

For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 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, M 2 /m≥M 1 /n does not hold always. In [K. Ch. Das, MATCH...

The first (M 1 ) and the second (M 2 ) Zagreb indices, as well as the first (M ¯ 1 ) and the second (M ¯ 2 ) Zagreb coindices, and the relations between them are examined. An upper bound on M 1 (T) and a lower bound on 2M 2 (T)+1 2M 1 (T) of trees is obtained, in terms of the number of vertices (n) and maximum degree (Δ). Moreover, we compare the Z...

Let G=(V,E) be a graph, d u the degree of its vertex u, and uv the edge connecting the vertices u and v. The atom-bond connectivity index and the sum-connectivity index of G are defined as ABC=∑ uv∈E (d u +d v -2)/(d u d v ) and χ=∑ uv∈E 1/d u +d v , respectively. Continuing the recent researches on ABC [B. Furtula, A. Graovac, D. Vukičević, Discr....

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M2(G)/m≥M1(G)/n, where M1 and M2 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...

Variable first and second Zagreb indices are defined by λ M 1 (G)=∑ v i ∈V d i 2λ and λ M 2 (G)=∑ u i v j ∈E d i λ ·d j λ , where d i is the degree of the vertex v i and λ is any real number. In this note, we obtain λ M 2 (G)≥ λ M 1 (G) for all unicyclic graphs and all λ∈[0,1].