The general Randić index Rα(G) is the sum of the weight d(u)d(v)α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α = 0, the first three minimum general Randić indices among trees are determined, and the corresponding extremal trees are characterized.

A topological index is a numeric quantity associated with chemical structure that attempts to link the various physicochemical properties, reactivity, or biological activity. Let R be commutative ring identity, and Z*(R) set of all non-zero zero divisors R. Then, Γ(R) said zero-divisor graph if only a·b=0, where a,b∈V(Γ(R))=Z*(R) (a,b)∈E(Γ(R)). We define a∼b a·b=0 a=b. ∼ always reflexive symmet...

Let Tn denote the set of all unrooted and unlabeled trees with n vertices, and (i, j) a double-star. By assuming that every tree of Tn is equally likely, we show that the limiting distribution of the number of occurrences of the double-star (i, j) in Tn is normal. Based on this result, we obtain the asymptotic value of Randić index for trees. Fajtlowicz conjectured that for any connected graph ...

Let G(n; m) be a connected graph without loops and multiple edges which has n vertices and m edges. We ÿnd the graphs on which the zeroth-order connectivity index, equal to the sum of degrees of vertices of G(n; m) raised to the power − 1 2 , attains maximum.

Let G be a simple connected graph and t be a given real number. The zero-order general Randić index αt(G) of G is defined as ∑ v∈V (G) d(v) t , where d(v) denotes the degree of v. In this paper, for any t , we characterize the graphs with the greatest and the smallest αt within two subclasses of connected unicyclic graphs on n vertices, namely, unicyclic graphs with k pendant vertices and unicy...

The Randić index R(G) of a graph G is defined as the sum of 1 √dudv over all edges uv of G, where du and dv are the degrees of vertices u and v, respectively. Let D(G) be the diameter of Gwhen G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path Pn achieves the minimum values for both R(G)/D(G) and R(G) − D(G). We prove this conjecture...

Introduction: Ensuring an adequate Depth of Anesthesia (DOA) during surgery is essential for anesthesiologists. Since the effect of anesthetic drugs is on the central nervous system, brain signals such as Electroencephalogram (EEG) can be used for DOA estimation. Anesthesia can interfere among brain regions, so the relationship among different areas can be a key factor in the anesthetic process...

BACKGROUND AND OBJECTIVES It is not well understood whether the left ventricular geometry is associated with such diastolic parameters as the left atrial volume and the left ventricular filling pressure, as assessed by the Doppler indices. Accordingly, this study aimed to evaluate the influence of the left ventricular geometry on the left atrial volume and the left ventricular filling pressure,...

Eccentric distance sum (EDS), which can predict biological and physical properties, is a topological index based on the eccentricity of a graph. In this paper we characterize the chain hexagonal cactus with the minimal and the maximal eccentric distance sum among all chain hexagonal cacti of length n, respectively. Moreover, we present exact formulas for EDS of two types of hexagonal cacti.

Let G be a simple connected graph of order n. The connectivity index Rα(G) of a graph G is the sum of the weights (d(u)d(v)) of all edges uv of G, where α is a real number (α 6= 0), and d(u) denotes the degree of the vertex u. In this paper, we present some new bounds for the connectivity index of a graph G in terms of the eigenvalues of the Laplacian matrix or adjacency matrix of the graph G, ...