Computational and analytical studies of the Randić index in Erdös-Rényi models
C. T. Martinez-Martinez, J. A. Mendez-Bermudez, Jose M. Rodriguez, Jose M. Sigarreta
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Computational and analytical studies of the Randi´cindex in Erd¨os-R´enyi models
C. T. Mart´ınez-Mart´ınez a , J. A. M´endez-Berm´udez a, ∗ , Jos´e M. Rodr´ıguez b ,Jos´e M. Sigarreta a,c a Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado PostalJ-48, Puebla 72570, Mexico b Universidad Carlos III de Madrid, Departamento de Matem´aticas, Avenida de laUniversidad 30, 28911 Legan´es, Madrid, Spain c Universidad Aut´onoma de Guerrero, Centro Acapulco CP 39610, Acapulco de Ju´arez,Guerrero, Mexico
Abstract
In this work we perform computational and analytical studies of theRandi´c index R ( G ) in Erd¨os-R´enyi models G ( n, p ) characterized by n ver-tices connected independently with probability p ∈ (0 , (cid:10) R ( G ) (cid:11) = h R ( G ) i / ( n/
2) scales with theproduct ξ ≈ np , so we can define three regimes: a regime of mostly isolatedvertices when ξ < .
01 ( R ( G ) ≈ . < ξ < < R ( G ) < n/ ξ > R ( G ) ≈ n/ (cid:10) R ( G ) (cid:11) , we analytically(i) obtain new relations connecting R ( G ) with other topological indices andcharacterize graphs which are extremal with respect to the relations obtainedand (ii) apply these results in order to obtain inequalities on R ( G ) for graphsin Erd¨os-R´enyi models. Keywords:
Randi´c index, vertex-degree-based topological index, randomgraphs, Erd¨os-R´enyi graphs. ∗ Corresponding author
Email address: [email protected] (J. A. M´endez-Berm´udez)
Preprint submitted to Applied Mathematics and Computation February 11, 2020 . Introduction
The interest in topological indices lies in the fact that they synthesizesome of the fundamental properties of a molecule into a single value. Withthis in mind, several topological indices have been studied so far; it is worthnoting the seminal work by Wiener (see [1]) in which he used the distancesof a chemical graph in order to model properties of alkanes.The
Randi´c connectivity index was defined in [2] as R ( G ) = X uv ∈ E ( G ) √ d u d v , (1)where uv denotes the edge of the graph G , and d u is the degree of the vertex u . Indeed, there are lots of works dealing with this index (see, e.g., [3, 4, 5]).In [6, 7, 8], the first and second variable Zagreb indices are defined as M α ( G ) = X u ∈ V ( G ) d αu , M α ( G ) = X uv ∈ E ( G ) ( d u d v ) α , with α ∈ R . The concept of variable molecular descriptors was proposedas a new way of characterizing heteroatoms in molecules (see [9, 10]). Theessential idea is that the variables are determined during the regression; thisallows to make the standard error of the estimate for a particular property(targeted in the study) as small as possible (see, e.g., [8]). The second variableZagreb index is used in the structure-boiling point modeling of benzenoidhydrocarbons [11].The general sum-connectivity index was defined in [12] as χ α ( G ) = X uv ∈ E ( G ) ( d u + d v ) α . Some relations of these indices are studied in ([13]).In addition to the multiple applications of the Randi´c index in physi-cal chemistry, this index has found several applications in other researchareas and topics, such as information theory [14], network similarity [15],protein alignment [16], network heterogeneity [17], and network robustness[18]. Moreover, in [19] the concept of graph entropy for weighted graphs wasintroduced, especially the Randi´c weights.2e want to recall that graphs have been widely used to study the prop-erties of highly complex systems. Among them we can mention biological,social, and technological networks [20, 21]. Moreover, graphs can be classi-fied as deterministic (regular and fractal) or disordered (random) [22]. De-terministic graphs follow specific construction rules, while in random graphsthe parameters take fixed values but the graph itself has a random structure.In the later case a statistical study of graph ensembles with the same averageproperties must be performed, since the analysis of a single random graph ismeaningless. There are well-known models of random graphs in the literature[23, 24], presumably the most popular are: the Erd¨os-R´enyi model of randomgraphs, scale-free networks (introduced by Barab´asi and Albert), and small-world networks (introduced by Watts and Strogatz). These three modelshave been extensively used to represent the organization of real-world com-plex systems (such as power grids or the Internet) through their underlyingnetwork structure [20, 23, 24].Although random graph models are not able to predict some propertiesobserved in real-world networks, such as nonvanishing clustering coefficientand power-law degree distributions [24], they have been deeply studied theo-retically (e.g. [25]). In fact, several important results, such as the emergenceof percolation, are analytically accesible from Erd¨os-R´enyi graphs [23, 25].Thus, here we consider Erd¨os-R´enyi random graphs, which were proposed bySolomonoff and Rapoport [26] and investigated later in great detail by Erd˝osand R´enyi [27, 28].This work is organized as follows. First, in Sec. 2 we perform a detailedscaling analysis of the average Randi´c index to find its universal parameter,i.e., the parameter that statistically fixes the average value of R ( G ). Then,in Sec. 3, we analytically (i) obtain new relations connecting R ( G ) with othertopological indices and (ii) apply these results in order to obtain inequalitieson R ( G ) for graphs in Erd¨os-R´enyi models.
2. Scaling analysis of the Randi´c index on Erd¨os-R´enyi graphs
We start with a computational (and statistical) study of the Randi´c indexon Erd¨os-R´enyi graphs. We consider random graphs G from the standardErd¨os-R´enyi model G ( n, p ), i.e., G has n vertices and each edge appearsindependently with probability p ∈ (0 , h R ( G ) i as a function ofthe probability p of Erd¨os-R´enyi graphs G ( n, p ) of several orders n . Here,the average h·i is computed over 2000 random graphs G ( n, p ). We observethat the curves of h R ( G ) i , for all the values of n considered here, have a verysimilar shape as a function of p : h R ( G ) i shows a smooth transition (in logscale) from zero to n/ p increases from zero (isolated vertices) to one(complete graphs). Note that n/ R ( G ) can take.Now, to ease our analysis, in Fig. 1(b) we present again h R ( G ) i but nownormalized to n/ (cid:10) R ( G ) (cid:11) = h R ( G ) i n/ . (2)From this figure we can clearly see that the main effect of increasing n is thedisplacement of the curves (cid:10) R ( G ) (cid:11) vs. p to the left on the p -axis. Moreover,the fact that these curves, plotted in semi-log scale, are shifted the sameamount on the p -axis when doubling n make us anticipate the existence ofa scaling parameter that depends on n . In order to search for that scalingparameter we first establish a measure to characterize the position of thecurves (cid:10) R ( G ) (cid:11) on the p -axis: We choose the value of p , that we label as p ∗ ,for which (cid:10) R ( G ) (cid:11) ≈ .
5; see the dashed line in Fig. 1(b). Notice that p ∗ locates the transition point from isolated vertices to complete Erd¨os-R´enyigraphs of size n .Then, in Fig. 2(a) we plot p ∗ versus n . The linear trend of the data (inlog-log scale) in Fig. 2(a) suggests the power-law p ∗ = C n δ . (3)In fact, Eq. (3) provides an excellent fitting to the data with C ≈ . δ ≈ −
1. Therefore, by plotting again the curves of (cid:10) R ( G ) (cid:11) now as afunction of the probability p divided by p ∗ , ξ ≡ pp ∗ ∝ pn δ ≈ pn − = np , (4)we observe that curves for different graph sizes n collapse on top of a single universal curve, see Fig. 2(b). This means that once the product np is fixed,the average Randi´c index on Erd¨os-R´enyi graphs is also fixed. This statementis in accordance with the results reported in [29, 30], where the spectral andtransport properties of Erd¨os-R´enyi graphs where shown to be universal forthe scaling parameter np , see also [31, 32, 33].4 .0001 0.001 0.01 0.1 1 p < R ( G ) > p < R ( G ) > n=25n=50n=100n=200n=400n=800 (a) (b) Figure 1: (a) Average Randi´c index h R ( G ) i as a function of the probability p of Erd¨os-R´enyi graphs G ( n, p ) of different sizes n ∈ [25 , h R ( G ) i normalized to n/ (cid:10) R ( G ) (cid:11) ,as a function of p . Dashed lines in (a) indicate the values of n/ n ∈ [200 , (cid:10) R ( G ) (cid:11) = 0 .
5, used to define p ∗ . Each symbol was computedby averaging over 2000 random graphs G ( n, p ).
10 100 1000 n p* ξ < R ( G ) > n=25n=50n=100n=200n=400n=800 (a) (b) Figure 2: (a) p ∗ (defined as the value of p for which (cid:10) R ( G ) (cid:11) ≈ .
5) as a function of thegraph size n . The red line is the fitting of Eq. (3) to the the data with fitting parameters C = 0 . δ = − . (cid:10) R ( G ) (cid:11) as a function of ξ . Vertical dashed linesin (b) indicate: The regime of mostly isolated vertices ( ξ < . . < ξ < ξ > .0001 0.01 1 p E n=25n=50n=100n=200n=400n=800 p E ξ E (a) (b) (c) Figure 3: (a) Randi´c Matrix energy E as a function of the probability p for Erd¨os-R´enyigraphs of size n . Dashed lines indicate the values of n/ n ∈ [200 , E = E/ ( n/
2) as a function p . (c) E as a function ξ . Additionally, from our previous experience, see e.g., [29, 30, 31, 32, 33],we expect that other quantities related to R ( G ) will also be scaled with ξ .Indeed, we validate this conjecture by analyzing the energy E ( n, p ) of theErd¨os-R´enyi graphs G ( n, p ) defined as [34, 35] E ( n, p ) = n X i =1 | e i | , (5)where e i are the eigenvalues of the corresponding Randi´c matrix [34, 35]: R ij = (cid:26) ( d i d j ) − / if v i ∼ v j , . (6)Thus in Fig. 3(a) we present the energy E as a function of the probability p of Erd¨os-R´enyi graphs of several sizes n . The curves E vs. p show a similarbehavior for different values of n : For small p , E increases with p until itreaches n/ E decreases from itsmaximum by further increasing p giving to the curves E vs. p a bell-likeshape in log scale. Now, for convenience, we normalize E to n/ E ) and plot it in Fig. 3(b). Here it is clear that the curves E vs. p arevery similar but shifted to the left on the p -axis for increasing n . Finally, inFig. 3(c) we plot E as a function of the scaling parameter ξ , see Eq. (4), andshow that all curves fall one on top of the other (except for finite size effectsat large ξ ). Therefore, we confirm that the energy of Erd¨os-R´enyi graphs (asdefined in Eq. (5)) also scales with the parameter ξ ; that is, once ξ is fixed6 R(G) P ( R ( G )) R(G)
R(G) n=25n=50n=100n=200n=400n=800 (a) (b) (c) ξ=0.298 ξ=0.754 ξ=1.666
Figure 4: Probability distribution functions of R ( G ), P ( R ( G )), for several graph sizes n at fixed values of (cid:10) R ( G ) (cid:11) : (a) (cid:10) R ( G ) (cid:11) = 0 .
25, (b) (cid:10) R ( G ) (cid:11) = 0 .
5, and (c) (cid:10) R ( G ) (cid:11) = 0 . ξ are given in the panels. Each histogram was constructedwith 2000 values of R ( G ). the normalized energy E is (statistically) the same for different parametercombinations ( n, p ). Additionally, from Fig. 3(c) we can conclude that themaximum value of E occurs in the interval 1 < ξ <
2, in close agreementwith the delocalization transition value for the eigenvectors of Erd¨os-R´enyigraphs reported in [29, 36, 37, 38, 39] to be ξ ≈ . ξ scales both (cid:10) R ( G ) (cid:11) and E reasonablywell, it is fair to say that there are additional quantities related to R ( G )which are still size dependent for fixed ξ . See for example Fig. 4, where weshow probability distribution functions of R ( G ) at fixed ξ . In this figure weobserve that, even for fixed ξ (or equivalently, for fixed (cid:10) R ( G ) (cid:11) ), P ( R ( G ))becomes narrower for increasing n . This means that the variance and theminimal and maximal values of R ( G ) change with n , as can be clearly seenin Fig. 5. This motivate us to look for bounds and inequalities on the Randi´cindex on Erd¨os-R´enyi graphs, which is the main topic of the following Section.
3. Inequalities for the Randi´c index on Erd¨os-R´enyi models
We recall that we consider a Random Graph G from the standard Erd¨os-R´enyi model G ( n, p ). In the following, G denotes a finite simple graph suchthat each connected component of G has, at least, one edge (there are noisolated vertices). We say that a statement holds for almost every graph ifthe probability of the set of graphs for which the statement fails tends to 0as n → ∞ . 7 .01 0.1 1 10 ξ v a r[ R ( G )] ξ m i n [ R ( G )] n=25n=50n=100n=200n=400n=800 ξ m a x [ R ( G )] (a) (c)(b) Figure 5: (a) var[ R ( G )], (b) min[ R ( G )], and (c) max[ R ( G )] as a function of ξ . Each symbolwas computed from 2000 values of R ( G ). The following facts about the Erd¨os-R´enyi model are well-known [40] (seealso [41]):(1) Almost every graph G has m = p n ( n − / o ( n ) edges.(2) Almost every graph G has maximum degree ∆ = p ( n − pqn log n ) / + o (( n log n ) / ), with p ∈ [1 / ,
1) and q = 1 − p .(3) Almost every graph G has minimum degree δ = q ( n − − (2 pqn log n ) / + o (( n log n ) / ), with p ∈ [1 / ,
1) and q = 1 − p .In the previous equalities we are using Landau’s notation: Recall that f ( n ) = g ( n ) + o ( a ( n )) means thatlim n →∞ f ( n ) − g ( n ) a ( n ) = 0 , and f ( n ) = g ( n ) + O ( a ( n )) means that f ( n ) − g ( n ) a ( n )is a bounded sequence.The following result relates the Randi´c and the ( − Theorem 1.
Let G be a graph with minimum degree δ and maximum degree ∆ . Then δ χ − ( G ) ≤ R ( G ) ≤ χ − ( G ) , if δ/ ∆ ≥ t , δ χ − ( G ) ≤ R ( G ) ≤ (∆ + δ ) √ ∆ δ χ − ( G ) , if δ/ ∆ < t , where t is the unique solution of the equation t + 5 t + 11 t − in theinterval (0 , . The equality in the lower bound is attained if and only if G isregular. The equality in the first upper bound is attained if and only if G isregular; the equality in the second upper bound is attained if and only if G isa biregular graph. Proof.
Since a ( t ) = t + 5 t + 11 t − , a (0) < a (1) >
0, there exists a unique solution of theequation t + 5 t + 11 t − , t is well-defined.Let us compute the maximum and minimum values of the function g :[ δ, ∆] × [ δ, ∆] → R given by g ( x, y ) = √ xy ( x + y ) . Since g ( x, y ) = g ( y, x ), we can assume that x ≤ y . The partial derivativesof g are ∂g∂x ( x, y ) = x − / y / ( x + y ) − x / y / x + y ) = x − / y / y − x x + y ) ,∂g∂y ( x, y ) = y − / x / x − y x + y ) . Since y ≥ x ≥ δ >
0, we obtain ∂g/∂y < g is a decreasing functionon y . Therefore, g attains its minimum value on { ( x, ∆) | δ ≤ x ≤ ∆ } andits maximum value on { ( x, x ) | δ ≤ x ≤ ∆ } . Note that g ( x, x ) = 1 / (4 x ) ≤ / (4 δ ).If ∆ ≤ δ , then ∂g/∂x ( x, ∆) < x > δ .If ∆ > δ , then ∂g/∂x ( x, ∆) > δ ≤ x < ∆ / ∂g/∂x ( x, ∆) < / < x ≤ ∆. 9ence, we have in every casemin n , √ ∆ δ (∆ + δ ) o = min (cid:8) g (∆ , ∆) , g ( δ, ∆) (cid:9) ≤ g ( x, y ) ≤ δ . Thus, min n , √ ∆ δ (∆ + δ ) o √ d u d v ≤ d u + d v ) ≤ δ √ d u d v , min n , √ ∆ δ (∆ + δ ) o R ( G ) ≤ χ − ( G ) ≤ δ R ( G ) . If the equality in the lower bound is attained, then ( d u , d v ) = ( δ, δ ) for all uv ∈ E ( G ); hence, d u = δ for all u ∈ V ( G ) and so, G is regular.In order to prove the upper bounds, it suffices to show that the inequality14∆ ≤ √ ∆ δ (∆ + δ ) (7)holds if and only if δ/ ∆ ≥ t .Inequality (7) is equivalent to the following statements(∆ + δ ) ≤ √ ∆ δ , (cid:16) δ ∆ (cid:17) ≤ r δ ∆ , (cid:16) δ ∆ (cid:17) ≤ δ ∆ , δ ∆ + 4 δ ∆ + 6 δ ∆ − δ ∆ + 1 ≤ . Since 0 < δ/ ∆ ≤
1, let us consider the function b ( t ) = t + 4 t + 6 t − t + 1for t ∈ (0 , b ( t ) = ( t − t + 5 t + 11 t −
1) = ( t − a ( t ), we have a ( t ) ≤ t ∈ [ t , δ/ ∆ ≥ t . Since the coefficients of the polynomial a ( t ) = t + 5 t + 11 t − t and t of the polynomial a ( t )are 1 and −
1, respectively, we have that t / ∈ Q . Note that this condition isequivalent to δ/ ∆ > t , since t / ∈ Q ; therefore, the equality in (7) is attainedif and only if δ = ∆.Therefore, the upper bounds hold.If δ/ ∆ ≥ t , then the previous argument gives that f attains its minimumvalue just at the point (∆ , ∆). Thus, the equality in the upper bound is10ttained if and only if ( d u , d v ) = (∆ , ∆) for every uv ∈ E ( G ), i.e., G isregular.If δ/ ∆ < t , then f attains its minimum value just at the points ( δ, ∆)and (∆ , δ ). Hence, the equality in the upper bound is attained if and only if { d u , d v } = { δ, ∆ } for every uv ∈ E ( G ), i.e., G is biregular. In this case, G can not be regular since δ < t ∆ < ∆.Theorem 1 have the following consequence on Random Graphs. Corollary 2.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q =1 − p , almost every graph G satisfies qn + O (( n log n ) / ) ≤ R ( G ) χ − ( G ) ≤ max n p , √ pq o n + O (( n log n ) / ) . Proof.
The conclusion in Theorem 1 can be written as follows:4 δ ≤ R ( G ) χ − ( G ) ≤ max n , (∆ + δ ) √ ∆ δ o . (8)Thus, the first inequality is a direct consequence of (8) and (3). Let us provethe second one. Items (2) and (3) give for almost every graph(∆ + δ ) √ ∆ δ = (cid:0) n + o (( n log n ) / ) (cid:1) p pqn + O ( n ( n log n ) / ) = n + o ( n ( n log n ) / ) √ pq n + O (( n log n ) / )= n √ pq (cid:16) − O (( n log n ) / ) √ pq n (cid:17) + o ( n ( n log n ) / ) √ pq n + O (( n log n ) / )= n √ pq + O (( n log n ) / ) + o (( n log n ) / )= n √ pq + O (( n log n ) / ) . This fact, (8) and item (2) give the second inequality for almost everygraph.Corollary 2 has the following consequence.
Corollary 3.
In the Erd¨os-R´enyi model G ( n, p ) , with p = 1 / , almost everygraph G satisfies R ( G ) χ − ( G ) = 2 n + O (( n log n ) / ) . Lemma 4.
Let g be the function g ( x, y ) = 2 √ xy/ ( x + y ) with < a ≤ x, y ≤ b . Then √ aba + b ≤ g ( x, y ) ≤ . Given a graph G , let us define δ G = min uv ∈ E ( G ) √ d u d v d u + d v , ∆ G = min uv ∈ E ( G ) √ d u d v d u + d v . Let G be a graph with maximum degree ∆ and minimum degree δ . ThenLemma 4 gives, for every uv ∈ E ( G ),2 √ ∆ δ ∆ + δ ≤ δ G ≤ √ d u d v d u + d v ≤ ∆ G ≤ . (9)Since δ G ≤ d u d v ( d u + d v ) √ d u d v ≤ ∆ G for every uv ∈ E ( G ), we obtain δ G d u + d v d u d v ≤ √ d u d v ≤ ∆ G d u + d v d u d v . (10)For every function f , we have X uv ∈ E ( G ) (cid:0) f ( d u ) + f ( d u ) (cid:1) = X u ∈ V ( G ) d u f ( d u ) , and so, X uv ∈ E ( G ) d u + d v d u d v = X uv ∈ E ( G ) (cid:16) d u + 1 d v (cid:17) = X u ∈ V ( G ) d u d u = X u ∈ V ( G ) n. This equality and (10) give the inequalities: nδ G ≤ R ( G ) ≤ n ∆ G .
12 similar result is proved in [41]; there, the author uses an argumentbased on differential calculus.As a consequence of the previous result and (9), we obtain the knowninequalities √ ∆ δ ∆ + δ n ≤ R ( G ) ≤ n . (11)Notice that the right inequality has already been computationally verified inFig. 1. Proposition 5.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q = 1 − p , almost every graph G satisfies R ( G ) ≥ √ pq n + O (( n log n ) / ) . Proof.
Let us consider the Erd¨os-R´enyi model G ( n, p ). Almost every graph G satisfies √ ∆ δ ∆ + δ n = p pqn + O ( n ( n log n ) / ) n + o (( n log n ) / ) n = √ pq n + O (( n log n ) / ) n + o (( n log n ) / ) n = √ pq n (cid:16) − o (( n log n ) / ) n (cid:17) + O (( n log n ) / ) n + o (( n log n ) / ) n = √ pq n + o (( n log n ) / ) + O (( n log n ) / )= √ pq n + O (( n log n ) / ) . This fact and (11) allow to obtain the result.
Corollary 6.
In the Erd¨os-R´enyi model G ( n, p ) , with p = 1 / , almost everygraph G satisfies R ( G ) = n O (( n log n ) / ) . In fact, this Corollary has already been computationally verified in Fig. 1.
Proposition 7.
Let G be a graph with n vertices, minimum degree δ andmaximum degree ∆ . Then n − δ (cid:0) M ( G ) − M / ( G ) (cid:1) ≤ R ( G ) ≤ n − (cid:0) M ( G ) − M / ( G ) (cid:1) , (cid:0) M ( G ) + 2 M / ( G ) (cid:1) − n ≤ R ( G ) ≤ δ (cid:0) M ( G ) + 2 M / ( G ) (cid:1) − n . The equality is attained in each bound if and only if G is a regular graph. roof. In the argument in the proof of [43, Theorem 1] appears the followingrelation: R ( G ) = n − X uv ∈ E ( G ) (cid:0) √ d u − √ d v (cid:1) d u d v , (12)and we deduce n − δ X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) ≤ R ( G ) ≤ n − X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) . Since X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) = X uv ∈ E ( G ) ( d u + d v ) − X uv ∈ E ( G ) p d u d v = M ( G ) − M / ( G ) , we obtain the first and second inequalities.Since − n + X uv ∈ E ( G ) (cid:0) √ d u + √ d v (cid:1) d u d v = − X u ∈ V ( G ) d u d u + X uv ∈ E ( G ) (cid:0) √ d u + √ d v (cid:1) d u d v = − X uv ∈ E ( G ) (cid:16) d u + 1 d v (cid:17) + X uv ∈ E ( G ) d u + d v + 2 √ d u d v d u d v = − X uv ∈ E ( G ) d u + d v d u d v + X uv ∈ E ( G ) d u + d v + 2 √ d u d v d u d v = X uv ∈ E ( G ) √ d u d v d u d v = 2 R ( G ) , we have R ( G ) = − n X uv ∈ E ( G ) (cid:0) √ d u + √ d v (cid:1) d u d v , − n X uv ∈ E ( G ) (cid:0)p d u + p d v (cid:1) ≤ R ( G ) ≤ − n δ X uv ∈ E ( G ) (cid:0)p d u + p d v (cid:1) . Since X uv ∈ E ( G ) (cid:0)p d u + p d v (cid:1) = X uv ∈ E ( G ) ( d u + d v )+2 X uv ∈ E ( G ) p d u d v = M ( G )+2 M / ( G ) ,
14e obtain the third and forth inequalities.If G is a regular graph, then δ = ∆ and, in each line, the lower and upperbounds are the same, and they are equal to R ( G ).If the equality is attained in some bound, then we have either d u d v = δ for every uv ∈ E ( G ) or d u d v = ∆ for every uv ∈ E ( G ). Thus, we haveeither d u = δ for every u ∈ V ( G ) or d u = ∆ for every u ∈ V ( G ), and so, thegraph is regular.Proposition 7 has the following consequence on random graphs. Corollary 8.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q =1 − p , almost every graph G satisfies q n + O ( n / (log n ) / ) ≤ M ( G ) − M / ( G ) n − R ( G ) ≤ p n + O ( n / (log n ) / ) ,q n + O ( n / (log n ) / ) ≤ M ( G ) + 2 M / ( G ) n + 2 R ( G ) ≤ p n + O ( n / (log n ) / ) . Proof.
Proposition 7 gives δ ≤ M ( G ) − M / ( G ) n − R ( G ) ≤ ∆ ,δ ≤ M ( G ) + 2 M / ( G ) n + 2 R ( G ) ≤ ∆ . Items (2) and (3) give for almost every graph∆ = (cid:0) pn + O (( n log n ) / ) (cid:1) = p n + O ( n / (log n ) / ) ,δ = (cid:0) qn + O (( n log n ) / ) (cid:1) = q n + O ( n / (log n ) / ) . These facts give the desired inequalities.The following proposition is a consequence of (12) in [43].
Proposition 9.
Let G be a graph with m edges, n vertices, minimum degree δ and maximum degree ∆ . Then then R ( G ) ≥ n − m (cid:16) √ δ − √ ∆ (cid:17) , and the equality is attained if and only if G is a regular or biregular graph. roof. Equation (12) can be written as R ( G ) = n − X uv ∈ E ( G ) (cid:16) √ d u − √ d v (cid:17) , and so, R ( G ) ≥ n − X uv ∈ E ( G ) (cid:16) √ δ − √ ∆ (cid:17) = n − m (cid:16) √ δ − √ ∆ (cid:17) . The equality is attained if and only if { d u , d v } = { δ, ∆ } for every uv ∈ E ( G ), i.e., G is a regular or biregular graph.Note that the lower bound in Proposition 9 is not comparable with theone in Corollary 5, as the following examples show:If G is the path graph with n vertices, then n − m (cid:16) √ δ − √ ∆ (cid:17) = n − n − (cid:16) − √ (cid:17) ≈ √ n is larger than √ ∆ δ ∆ + δ n = √ n, for large enough n . However, if G is the complete graph with n − K n − with an additional edge joining a vertex of K n − with an additionalvertex of degree 1, then √ ∆ δ ∆ + δ n = √ n − n n = √ n − n − m (cid:16) √ δ − √ ∆ (cid:17) = n − ( n − n −
2) + 12 (cid:16) − √ n − (cid:17) , for large enough n . Proposition 10.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q = 1 − p , almost every graph G satisfies R ( G ) ≥ √ pq + 2 q − q n + o ( n ) , if p > / ,R ( G ) = n O (( n log n ) / ) , if p = 1 / . roof. The second statement follows from Corollary 6.Assume now p > /
2. Items (2) and (3) give that in the Erd¨os-R´enyimodel G ( n, p ), almost every graph G satisfies (cid:16) √ δ − √ ∆ (cid:17) = ∆ + δ − √ ∆ δ ∆ δ = n + o (( n log n ) / ) − p pqn + O ( n ( n log n ) / ) pqn + O ( n ( n log n ) / )= n + o (( n log n ) / ) − √ pq n + O (( n log n ) / ) pqn + O ( n ( n log n ) / )= (cid:0) − √ pq (cid:1) n + O (( n log n ) / ) pqn + O ( n ( n log n ) / )= 1 − √ pqpqn + o (cid:16) n (cid:17) . This fact, Proposition 9 and item (1) give R ( G ) ≥ n − m (cid:16) √ δ − √ ∆ (cid:17) = n − (cid:16) pn ( n − o (cid:0) n (cid:1)(cid:17)(cid:16) − √ pqpqn + o (cid:16) n (cid:17)(cid:17) = 12 n − − √ pq q n + o ( n ) = 2 √ pq + 2 q − q n + o ( n ) . The misbalance rodeg index is defined as MR ( G ) = X uv ∈ E ( G ) (cid:12)(cid:12)p d u − p d v (cid:12)(cid:12) . This is a significant predictor of enthalpy of vaporization and of standardenthalpy of vaporization for octane isomers (see [44]).
Theorem 11.
Let G be a graph with maximum degree ∆ and m edges. Then R ( G ) ≤ n − m MR ( G ) , and the equality is attained if and only if G is regular. roof. By Cauchy-Schwarz inequality we have MR ( G ) = (cid:16) X uv ∈ E ( G ) (cid:12)(cid:12)p d u − p d v (cid:12)(cid:12)(cid:17) ≤ (cid:16) X uv ∈ E ( G ) (cid:17) X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) ≤ m X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) . Hence, (12) gives R ( G ) = n − X uv ∈ E ( G ) (cid:0) √ d u − √ d v (cid:1) d u d v ≤ n − X uv ∈ E ( G ) (cid:0)p d u − p d v (cid:1) ≤ n − m MR ( G ) . If G is regular, then R ( G ) = n/ MR ( G ) = 0 and so, the equality isattained.If the equality is attained, then d u d v = ∆ for every uv ∈ E ( G ); thus, d u = ∆ for all u ∈ V ( G ) and so, G is a regular graph. Corollary 12.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q =1 − p , almost every graph G satisfies MR ( G ) n − R ( G ) ≤ p n + o ( n ) . Proof.
Theorem 11 gives the inequality MR ( G ) n − R ( G ) ≤ ∆ m. Items (1) and (2) give that in the Erd¨os-R´enyi model G ( n, p ), almost everygraph G satisfies∆ m = (cid:0) pn + O (( n log n ) / ) (cid:1) (cid:16) pn ( n − o (cid:0) n (cid:1)(cid:17) = 12 p n + o ( n ) , and this gives the desired inequality.The following Sz¨okefalvi Nagy inequality appears in [45] (see also [46]).18 emma 13. If a j ≥ for ≤ j ≤ k , R = max j a j and r = min j a j , then k k X j =1 a j − (cid:16) k X j =1 a j (cid:17) ≥ k R − r ) . In many papers the hypothesis a j ≥ ≤ j ≤ k , R = max j a j and r = min j a j , is replaced by 0 < r ≤ a j ≤ R for 1 ≤ j ≤ k . However,the conclusion of Lemma 13 does not hold in general with the hypothesis0 < r ≤ a j ≤ R for 1 ≤ j ≤ k , as the following example shows:If a j = a for 1 ≤ j ≤ k , R > a and r ≤ a < R , then k k X j =1 a j − (cid:16) k X j =1 a j (cid:17) = k a − k a = 0 < k R − r ) . Theorem 14.
Let G be a graph with m edges, Π = max uv ∈ E ( G ) √ d u d v , and π = min uv ∈ E ( G ) √ d u d v . Then R ( G ) ≤ r mM − ( G ) − m − π ) , and the equality is attained if G is a regular or biregular graph. Proof.
If we choose a j = 1 / √ d u d v , Lemma 13 gives mM − ( G ) − R ( G ) = m X uv ∈ E ( G ) d u d v − X uv ∈ E ( G ) √ d u d v ≥ m − π ) , and this gives the inequality.If G is a biregular or regular graph, then1 √ d u d v = 1 √ ∆ δ = Π = π for every uv ∈ E ( G ). Thus, r mM − ( G ) − m P − p ) = r m m ∆ δ = m √ ∆ δ = R ( G ) . inverse degree index ID ( G ) is defined by ID ( G ) = X u ∈ V ( G ) d u = X uv ∈ E ( G ) (cid:16) d u + 1 d v (cid:17) = X uv ∈ E ( G ) d u + d v d u d v . The inverse degree index of a graph has been studied by several authors (see,e.g., [47, 48, 49] and the references therein). The following result providessome inequalities relating Randi´c and Inverse Degree indices (see [50] forother inequalities relating these indices).
Theorem 15.
Let G be a graph with minimum degree δ and maximum degree ∆ . Then δ ID ( G ) ≤ R ( G ) ≤ ∆2 ID ( G ) , if δ ≥ s ∆ , (∆ δ ) / ∆ + δ ID ( G ) ≤ R ( G ) ≤ ∆2 ID ( G ) , if δ ≤ s ∆ , where s is the unique solution of the equation s − √ s + 1 = 0 in (0 , .Furthermore, the upper bound is attained if and only if G is regular; if δ ≥ s ∆ , then the lower bound is attained if and only if G is regular; if δ ≤ s ∆ ,then the lower bound is attained if and only if G is biregular. Proof.
First of all, let us check that s is well-defined, i.e., there existsa unique solution of the equation s − √ s + 1 = 0 in (0 , s = t , we see that this holds if and only if thereexists a unique solution of the equation t − t + 1 = 0 in (0 , t − t + 1 = ( t − u ( t ), with u ( t ) = t + t + t −
1. Since u (0) = − u (1) = 2and u ′ ( t ) = 3 t + 2 t + 1 > , t of u in (0 ,
1) and, in fact, u ( t ) < t ∈ (0 , t ) and u ( t ) > t ∈ ( t , s = t , then s − √ s + 1 > s ∈ (0 , s ) and s − √ s + 1 < s ∈ ( s , f : [ δ, ∆] × [ δ, ∆] → R be the function given by f ( x, y ) = (cid:16) x + 1 y (cid:17) √ xy = x − / y / + y − / x / . f . We can assumethat x ≤ y (symmetry). ∂f∂x ( x, y ) = − x − / y / + 12 x − / y − / = 12 x − / y − / ( x − y ) . Thus, ∂f∂x ( x, y ) < , if δ ≤ x ≤ y ≤ ∆ , and so, the function f attains its maximum value in the set { x = δ, δ ≤ y ≤ ∆ } , and the minimum value in the set { δ ≤ x = y ≤ ∆ } . Thus, f ( x, y ) ≥ min δ ≤ x ≤ ∆ f ( x, x ) = min δ ≤ x ≤ ∆ x x = 2∆ , d u + 1 d v ≥
2∆ 1 √ d u d v ,R ( G ) ≤ ∆2 ID ( G ) . Since ∂f∂y ( x, y ) = 12 y − / x − / ( y − x ) , if ∆ − δ <
0, then ∂f∂y ( δ, y ) = 12 y − / δ − / ( y − δ ) ≤ y − / δ − / (∆ − δ ) < , and f ( x, y ) ≤ max δ ≤ y ≤ ∆ f ( δ, y ) = f ( δ, δ ) = 2 δ . If ∆ − δ ≥
0, then ∂f∂y ( δ, y ) = 12 y − / δ − / ( y − δ ) ≤ y ∈ [ δ, √ δ ]. Thus, f ( δ, y ) decreases on [ δ, √ δ ] and increaseson [ √ δ, ∆]. Hence, we have in both cases f ( x, y ) ≤ max δ ≤ y ≤ ∆ f ( δ, y ) = max (cid:8) f ( δ, δ ) , f ( δ, ∆) (cid:9) = max n δ , (cid:16) δ + 1∆ (cid:17) √ δ ∆ o . s − √ s + 1 > , s ). Thus, we have for δ ≤ s ∆, (cid:16) δ ∆ (cid:17) ≥ r δ ∆ , (cid:16) δ + 1∆ (cid:17) √ δ ∆ ≥ δ , and we conclude f ( x, y ) ≤ max n δ , (cid:16) δ + 1∆ (cid:17) √ δ ∆ o = ∆ + δ (∆ δ ) / , d u + 1 d v ≤ ∆ + δ (∆ δ ) / √ d u d v ,R ( G ) ≥ (∆ δ ) / ∆ + δ ID ( G ) . If δ ≥ s ∆, then f ( x, y ) ≤ f ( δ, δ ) = 2 /δ and R ( G ) ≥ δ ID ( G ) . The previous argument gives that the upper bound is attained if and onlyif d u = d v = ∆ for every uv ∈ E ( G ), and this happens if and only if G isregular.Assume that δ ≥ s ∆. Thus, the lower bound is attained if and only if d u = d v = δ for every uv ∈ E ( G ), i.e., if and only if G is regular.Assume that δ ≤ s ∆. Thus, the lower bound is attained if and only if { d u , d v } = { ∆ , δ } for every uv ∈ E ( G ), i.e., if and only if G is biregular (notethat G can not be a regular graph since δ ≤ s ∆ < ∆).Theorem 15 has the following consequence on random graphs. Corollary 16.
In the Erd¨os-R´enyi model G ( n, p ) , with p ∈ [1 / , and q =1 − p , almost every graph G satisfies min n q , ( pq ) / p + q o n + O (( n log n ) / ) ≤ R ( G ) ID ( G ) ≤ p n + O (( n log n ) / ) . Proof.
Theorem 15 can be stated as follows:min n δ , (∆ δ ) / ∆ + δ o ≤ R ( G ) ID ( G ) ≤ ∆2 . δ ) / ∆ + δ = (cid:0) pqn + O ( n ( n log n ) / ) (cid:1) / ( p + q ) n + O ( n ( n log n ) / ) = ( pq ) / n (cid:0) O ( n ( n log n ) / ) pqn (cid:1) ( p + q ) n + O ( n ( n log n ) / )= ( pq ) / n ( p + q ) n + O ( n ( n log n ) / ) + O ( n ( n log n ) / )( p + q ) n + O ( n ( n log n ) / )= ( pq ) / np + q (cid:16) − O (( n log n ) / ) n (cid:17) + O (( n log n ) / )= ( pq ) / p + q n + O (( n log n ) / ) . These facts, and items (2) and (3) give for almost every graphmin n q n + O (( n log n ) / ) , ( pq ) / p + q n + O (( n log n ) / ) o ≤ H ( G ) ID ( G ) ≤ p n + O (( n log n ) / ) , and this finishes the proof.
4. Summary
Based on the important theoretical-practical applications of the Randi´cindex, in this paper we have studied computationally and analytically theproperties of the Randi´c index R ( G ) in Erd¨os-R´enyi graphs G ( n, p ) charac-terized by n vertices connected independently with probability p ∈ (0 , (cid:10) R ( G ) (cid:11) = h R ( G ) i / ( n/ ξ ≈ np is the scal-ing parameter of R ( G ( n, p )); that is, for fixed ξ , (cid:10) R ( G ) (cid:11) is also fixed, seeFig. 2(b). Moreover, our analysis provides a way to predict the value ofthe Randi´c index on Erd¨os-R´enyi graphs once the value of ξ is known: R ( G ) ≈ ξ < .
01 (when the vertices in the graph are mostly iso-lated), the transition from isolated vertices to complete graphs occurs in theinterval 0 . < ξ <
10 where 0 < R ( G ) < n/
2, while when ξ >
10 the graphsare almost complete and R ( G ) ≈ n/
2. These intervals are indicated as verti-cal dashed lines in Fig. 2(b). Also, to extend the applicability of our scalinganalysis we demonstrate that for fixed ξ the spectral properties of R ( G ( n, p ))(characterized by the energy of the corresponding Randi´c matrix) are also23niversal; i.e., they do not depend on the specific values of the individualgraph parameters, see Fig. 3(c).In particular, we would like to stress that here we have successfully in-troduced a scaling approach to the study of topological indexes.Then, to complement the study of the Randi´c index we have exploredthe relations between R ( G ) and other important topological indexes suchas the (-2) sum-connectivity index, the misbalance rodeg index, the inversedegree index, among others. In particular, we characterized graphs whichare extremal with respect to those relations. Acknowledgements
C.T.M.-M. and J.A.M.-B. thank partial support by VIEP-BUAP (GrantNo. MEBJ-EXC18-G), Fondo Institucional PIFCA (Grant No. BUAP-CA-169), and CONACyT (Grant No. CB-2013/220624), Mexico. J.M.R. andJ.M.S. were supported in part by two grants from Ministerio de Econom´ıa yCompetitividad, Agencia Estatal de Investigaci´øn (AEI) and Fondo Europeode Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain.
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