scholarly journals On some mathematical properties of the general zeroth-order Randić coindex of graphs

2020 ◽  
Vol 12 (2) ◽  
pp. 75-82
Author(s):  
I. Milovanović ◽  
M. Matejić ◽  
E. Milovanović ◽  
A. Ali
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Degree Sequence ◽  
Tree Structure ◽  
Vertex Degree ◽  
Arbitrary Real Number ◽  
Zeroth Order ◽  
Mathematical Properties ◽  
Simple Connected Graph

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1 i + d a-1 j ) = ∑ n i=1 (n-1-di)d a-1 i , where a is an arbitrary real number. Similarly, general zerothorder Randic coindex of ' G is defined as 0Ra(G) = ∑ n i=1 di(n-1-di) a-1 . New lower bounds for 0Ra(G) and 0Ra(G) are obtained. A case when G has a tree structure is also covered.

scholarly journals Some inequalities for general zeroth-order Randic index

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5249-5258
Author(s):  
Predrag Milosevic ◽  
Igor Milovanovic ◽  
Emina Milovanovic ◽  
Marjan Matejic
Keyword(s):  
Connected Graph ◽  
Degree Sequence ◽  
Vertex Degree ◽  
Arbitrary Real Number ◽  
Randić Index ◽  
Zeroth Order ◽  
Real Numbers ◽  
Randic Index ◽  
Simple Connected Graph ◽  
New Inequalities

Let G=(V,E), V={v1, v2,..., vn}, be a simple connected graph with n vertices, m edges and vertex degree sequence ? = d1?d2 ?...? dn = ? > 0, di = d(vi). General zeroth-order Randic index of G is defined as 0R?(G) = ?ni =1 d?i , where ? is an arbitrary real number. In this paper we establish relationships between 0R?(G) and 0R?-1(G) and obtain new bounds for 0R?(G). Also, we determine relationship between 0R?(G), 0R?(G) and 0R2?-?(G), where ? and ? are arbitrary real numbers. By the appropriate choice of parameters ? and ?, a number of old/new inequalities for different vertex-degree-based topological indices are obtained.

scholarly journals Some remarks on general sum-connectivity coindex

2020 ◽  
Vol 12 (1) ◽  
pp. 29-35
Author(s):  
M.M. Matejić ◽  
E.I. Milovanović ◽  
I. Milovanović
Keyword(s):  
Real Number ◽  
Connected Graph ◽  
Arbitrary Real Number ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G = (V,E), V = {v1, v2,..., vn} be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d1 ≥ d2 ≥ ··· ≥ dn > 0, di = d(vi). The general sumconnectivity coindex is defined as Ha(G) = ∑i j (di + dj) a , while multiplicative first Zagreb coindex is defined as P1(G) = ∏i j (di + dj). Here a is an arbitrary real number, and i j denotes that vertices i and j are not adjacent. Some relations between Ha(G) and P1(G) are obtained.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

scholarly journals Bounds for symmetric division deg index of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 683-698 ◽  
Author(s):  
Kinkar Das ◽  
Marjan Matejic ◽  
Emina Milovanovic ◽  
Igor Milovanovic
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Topological Index ◽  
Minimum Degree ◽  
Topological Indices ◽  
Symmetric Division ◽  
Mathematical Properties ◽  
Vertex Degrees ◽  
Additive Modeling ◽  
Simple Connected Graph

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

2017 ◽  
Vol 97 (1) ◽  
pp. 1-10
Author(s):  
I. MILOVANOVIĆ ◽  
M. MATEJIĆ ◽  
E. GLOGIĆ ◽  
E. MILOVANOVIĆ
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Its Sequence ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let$G$be a simple connected graph with$n$vertices and$m$edges and$d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$its sequence of vertex degrees. If$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$are the Laplacian eigenvalues of$G$, then the Kirchhoff index of$G$is$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for$\mathit{Kf}(G)$in terms of some of the parameters$\unicode[STIX]{x1D6E5}=d_{1}$,$\unicode[STIX]{x1D6E5}_{2}=d_{2}$,$\unicode[STIX]{x1D6E5}_{3}=d_{3}$,$\unicode[STIX]{x1D6FF}=d_{n}$,$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$and the topological index$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.

scholarly journals Relations between ordinary and multiplicative degree-based topological indices

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 3031-3042 ◽  
Author(s):  
Ivan Gutman ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

scholarly journals Lower Bounds for Inverse Sum Indeg Index of Graphs

2020 ◽  
Vol 44 (4) ◽  
pp. 551-562
Author(s):  
Ivan Gutman ◽  
M. MATEJIC ◽  
E. MILOVANOVIC ◽  
I. MILOVANOVIC
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G = (V,E), V = {1, 2,…,n}, be a simple connected graph with n vertices and m edges and let d1 ≥ d2 ≥⋅ ⋅⋅≥ dn > 0, be the sequence of its vertex degrees. With i ∼ j we denote the adjacency of the vertices i and j in G. The inverse sum indeg index is defined as ISI = ∑ -didj- di+dj with summation going over all pairs of adjacent vertices. We consider lower bounds for ISI. We first analyze some lower bounds reported in the literature. Then we determine some new lower bounds.

A note on the Laplacian resolvent energy of graphs

2019 ◽  
Vol 13 (06) ◽  
pp. 2050119
Author(s):  
M. Matejić ◽  
E. Zogić ◽  
E. Milovanović ◽  
I. Milovanović
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Laplacian Eigenvalues ◽  
Energy Of Graph ◽  
Simple Connected Graph

Let [Formula: see text] be a simple connected graph with [Formula: see text] vertices, [Formula: see text] edges and let [Formula: see text] be its Laplacian eigenvalues. The Laplacian resolvent energy of graph [Formula: see text] is defined by [Formula: see text]. In this paper, we give some new lower bounds for the invariant [Formula: see text].

scholarly journals Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

2021 ◽  
Vol 7 (2) ◽  
pp. 2529-2542
Author(s):  
Chang Liu ◽  
◽  
Jianping Li
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Randić Index ◽  
Extremal Graphs ◽  
Zeroth Order ◽  
Randic Index ◽  
General Randić Index ◽  
General Randic Index

<abstract><p>The zeroth-order general Randić index of graph $ G = (V_G, E_G) $, denoted by $ ^0R_{\alpha}(G) $, is the sum of items $ (d_{v})^{\alpha} $ over all vertices $ v\in V_G $, where $ \alpha $ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $ ^0R_{\alpha} $ of trees with a given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.</p></abstract>

scholarly journals Lower and Upper Bounds for Hyper-Zagreb Index of Graphs

2020 ◽  
pp. 1401-1406
Author(s):  
G. H. SHIRDEL ◽  
H. REZAPOUR ◽  
R. NASIRI
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Topological Indices ◽  
Upper And Lower Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Simple Connected Graph

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 .

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