Some Less Familiar Properties of Randić Index

Several mathematically relevant properties of the Randić connectivity index, that may be less familiar to the chemical community, are outlined and commented.


INTRODUCTION
N his seminal and thousands-times quoted paper [1] "On characterization of molecular branching", Milan Randić introduced one of the most popular graph-based structure descriptors (topological indices). Randić himself named it "branching index", but the name was soon changed into "connectivity index". [2,3] Eventually, the name "Randić index" (or, sometimes, "Randić connectivity index") has been accepted by the vast majority of contemporary scholars. It should be noted that "Randić index" is explicitly mentioned in the newest (2020) Mathematics Subject Classification, within the section 05 Combinatorics, subsection 05C Graph theory, and subject 05C09 Graphical indices; for details see. [4] Let G be a simple (molecular) graph with n vertices and m edges, whose vertex set is 1 2 and edge set ( ) E G . The edge connecting the vertices i v and j v is denoted by ij . The degree (= number of first neighbors) of the vertex i v will be denoted by i d . In order to avoid trivialities, in what follows it is assumed that the graph G is connected, possessing at least two vertices (and therefore at least one edge). Thus, G has no isolated vertices, i.e., 1 i d ≥ for all 1,2,..., i n = . For additional notions from chemical graph theory, the textbooks [5,6] should be consulted.
In the above specified notation, the Randić index is defined as [1] ( ) It should be noted that the Randić index is one of the several vertex-degree-based molecular structure descriptors whose general form is where ( , ) F x y is a suitably chosen function whose property must be ( , ) ( , ) F x y F y x = . Nowadays, a few dozens of such topological indices are studied in the current mathematical and chemical literature. [7][8][9][10][11]

GENERALIZATIONS OF RANDIĆ INDEX
Since 1975, when the Randić index was introduced, its numerous generalizations were considered and investigated. The most straightforward is of the form for α being some real number. This expression is based on the fact that the ordinary Randić index, Eq. (1), pertains to 1 / 2 α = − . α R is usually referred to as the "general Randić index". [12,13] The case 1 α = − attracted special attention of mathematicians. [14,15] The expression on the right-hand side of Eq. (1) can be viewed as a connectivity index of the first order. If so, then the "second-order connectivity index" would be [2,3] in spite of their simplicity, have also been much investigated. [12,13] In what follows, we focus our attention only to the mathematical properties of the ordinary Randić index, Eq. (1).

AN IDENTITY FOR RANDIĆ INDEX
The remarkable identity (2) was communicated in two papers, [16,17] whereas its main chemical applications in the paper [18] and elsewhere. [19] From Theorem 1, the following is immediately evident: Corollary 3. The equality ( ) / 2 R G n = holds for any regular graph with n vertices. If the (molecular) graph G is not regular, then ( ) / 2 R G n < .
The results of Corollaries 2 and 3 were discovered in the 1980s, [19] but were proven without the use of formula (2). Corollary 3 has a noteworthy chemical consequence: in the case of regular graphs (such as the molecular graphs of fullerenes, nanotubes, and similar), the Randić index is completely insensitive to any structural detail (except the number of carbon atoms). Therefore, the applicability of Randić index is limited to (molecular) graphs in which the vertex degrees assume different values. Trees (molecular graphs of alkanes) are the best objects satisfying this condition. Indeed, the majority of applications of the Randić index is for (molecular) trees. The fundamental result along these lines is the following: Theorem 4. Let n S and n P denote the n-vertex star and path (see Figure 1). Then for any n-vertex tree n T , different from n S and n P , Since, evidently, the path is the least branched and the star the most branched tree, inequalities (3) confirm that the Randić index is a proper measure of molecular branching.
In order to see how Theorem 4 can be deduced from Eq. (2), consider the term and note that in the case of trees, its maximal value is for 1 , 1 = or vice versa. All the 1 n − edges of n S assume this maximal value. Therefore n S has the greatest Randić index among n-vertex trees.
A path has two pendent edges (= edges incident to a vertex of degree 1). Any other tree has 3 or more pendent edges.
The smallest value of ij X is for i j d d = , equal to zero. In the case of pendent edges, the second-smallest is 1.
or vice versa. The path n P has 3 n − edges for which 0 ij X = and two (= the smallest possible number) with 1.5 2 ij X = − . This evidently renders the smallest Randić index among n-vertex trees.
It is worth noting that the proof of the left-hand side of inequality (3), without using the identity (2), was the first non-trivial mathematical result obtained for the Randić index. [14] The publication of the papers [14,15] triggered intensive mathematical research of the Randić index and its generalizations, resulting in hundreds of publications and at least two books. [12,13]

RANDIĆ MATRIX
There are numerous matrices that can be associated with graphs. [21,22] However, only two graph matrices have found extensive applications in both mathematics, computer science, and natural sciences. These are the adjacency matrix ( ) A G and the Laplacian matrix ( ) L G . Both are symmetric square matrices of order n. Their ( , ) i j -elements are defined as Croat. Chem. Acta 2020, 93(4) Details on the above defined graph matrices can be found in the monographs. [23,24] For what follows, we need to define an auxiliary matrix ( ) D D G = whose diagonal elements are the vertex degrees 1 2 , ,..., n d d d , whereas all off-diagonal elements are zero. Then the Laplacian and adjacency matrices are related as The normalized Laplacian matrix is defined as [25] 1/2 where n I stands for the unit matrix of order n. By direct calculation we find that the ( , ) i j -element of the matrix 1/2 Bearing in mind Eq. (1), we arrive at the following definitions. [26,27] Definition 5. The matrix whose ( , ) i j -element is given by Eq. (6) is the Randić matrix of the underlying graph G and will be denoted by ( ) RM RM G = .
Thus, ( ) RM G is a symmetric square matrix of order n, with zero diagonal, whose sum of all elements is equal to twice the Randić index of the graph G. Definition 6. The eigenvalues of the Randić matrix, denoted by 1 2 , ,..., n ρ ρ ρ , are the Randić eigenvalues of the graph G. They form the Randić spectrum of the graph G. The greatest Randić eigenvalue is said to be the Randić spectral radius of the graph G.
The Randić eigenvalues are real-valued numbers, whose sum is equal to zero.
is the Randić energy of the graph G.
The above Definitions 5-7 were first time stated in the papers, [26,27] and were introduced knowing Eq. (5). It is worth noting that the paper [27] was dedicated to Milan Randić, on the occasion of his 80 th birthday.
By recognizing Eq. (5), it became evident that the concept of Randić matrix (and, indirectly, of the Randić index) has a deep-lying algebraic root, and is connected with the theory of Laplacian matrices.
Since 2010, when the papers [26,27] were published, the concept of Randić matrix, and especially of its spectrum and energy, attracted much attention and were studied in scores of publications. Some of these are surveyed in the subsequent section.

RANDIĆ ENERGY
If the eigenvalues of the adjacency matrix ( ) A G are 1 2 , ,..., n λ λ λ , then the energy of the graph G is defined as [28] The relation between graph energy and the total π-electron energy is described in detail elsewhere. [28,29] The graph energy showed to be a significant and fruitful mathematical concept and its research resulted in hundreds of results and publications. [30,31] This success was a motivation to introduce energies of other matrices. [29] One among these is the Randić energy, Eq. (7), Definition 7. Evidently, Eq. (7) is constructed in full analogy with Eq. (8).
In the same year (2010) when the Randić energy was introduced, [26,27] Cavers et al. [32] put forward the concept of normalized Laplacian energy, namely the energy of the normalized Laplacian matrix It was not long to recognize that the two energies coincide. [26] Theorem 8. For any connected graph G, the normalized Laplacian matrix and the Randić matrix are related as ( ) ( ) In the literature that followed, there were a few publications concerned with the normalized Laplacian energy, e.g. the recent papers, [33][34][35] but the majority of them use the name "Randić energy". According to our census, [30,31] by 2020, there are 30 publications whose titles mention Randić energy, plus a review. [36] The fundamental result on Randić energy, analogous to Theorem 4 is: [32,37,38] Claim 10. (a) The n-vertex tree with the smallest Randić energy is the sun-graph (if n is odd) and the balanced double sun (if n is even), see Figure 2.
(b) The n-vertex tree with the greatest Randić energy is the star, see Figure 1. This is also the n-vertex graph with greatest Randić energy.
Claim 10(b) was proven by Cavers et al. [32] Claim 10(a) was established in [37,38] by means of computer search, but its rigorous mathematical proof is still missing.
Most papers dealing with Randić energy report lower and upper bounds and/or establish (or conjecture) the extremal cases. A characteristic result of this kind is the following: [39] Theorem 11. Let G be a connected graph with n vertices. Let 1 R − be the general Randić index for 1 α = − . Then If the maximum degree of G is equal to n-1, then both equalities hold if and only if Note that the determinants of the Randić and adjacency matrices are related as 1 2 det ( ) det ( ) and recall that the structure-dependency of det ( ) A G of molecular graphs is studied in full detail. [40] In a few papers, [41,42] Randić energies of special graphs were computed.

Theorem 12.
Let G be a connected graph whose maximal vertex degree is Δ . Then [57,58] Equality on the left-hand side is attained only if 2 G P ≅ . Equality on the right-hand side is attained only for , a b G K ≅ , a complete bipartite graph.
The left inequality was discovered by Yan et al. [57] whereas the right inequality by Gerardo and Octavio Arizmendi. [58] In spite of their simplicity, proving the inequalities (9) is a very difficult mathematical task. Namely, whereas ( ) R G is a vertex-degree-dependent quantity, ( ) GE G depends on graph eigenvalues. Thus, (9) connect two mathematically fully unrelated structure descriptors.
From the inequalities (9), one could expect that Randić index and graph energy are linearly correlated. This, however, is not the case. Numerical testing on trees and benzenoid systems revealed that the relation between ( ) R G and ( ) GE G is much more complex than linear. The most convincing example for the absence of any correlation between ( ) R G and ( ) GE G are regular graphs (e.g., molecular graphs of fullerenes). For such graphs, according to Corollary 3, the Randić index is a constant, independent of any variation in (molecular) structure. In contrast to this, graph energy of regular graphs has a pronounced sensitivity to their structure. [28,59,60]

CONCLUDING REMARKS
By reading Milan Randić's original paper, [1] our impression is that the author was not aware of the deep-lying mathematical roots of his "branching index". Such fortunate Figure 2. The trees whose Randić energy is minimal. Here the sun-graph with n = 11 vertices and the balanced double sun with n = 22 vertices are depicted. Note that these differ very much from the path (Figure 1), implying that the Randić energy is not a measure of molecular branching. In the case of even n, the degrees of the two central vertices of the double sun are either equal or differ by one.