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

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.

Some remarks on general sum-connectivity coindex

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.

F-hypercyclic operators on Fréchet spaces

We investigate F-hypercyclicity of linear, not necessarily continuous, operators on Frechet spaces. The notion of lower (mn)-hypercyclicity seems to be new and not considered elsewhere even for linear continuous operators acting on Frechet spaces. We pay special attention to the study of q-frequent hypercyclicity, where q > 1 is an arbitrary real number. We present several new concepts and results for lower and upper densities in a separate section, providing also a great number of illustrative examples and open problems.

Some inequalities for general zeroth-order Randic index

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.

Vertex Connectivity of Fuzzy Graphs with Applications to Human Trafficking

Connectivity is the most important aspect of a dynamic network. It has been widely studied and applied in different perspectives in the past. In this paper, constructions of [Formula: see text]-connected fuzzy graphs for an arbitrary real number [Formula: see text] and average fuzzy vertex connectivity of fuzzy graphs are discussed. Average fuzzy vertex connectivity of fuzzy trees, fuzzy cycles and complete fuzzy graphs are studied. The concept of a uniformly [Formula: see text]-connected fuzzy graph is introduced and characterized towards the end. An application related to human trafficking is also discussed.

A problem of coefficient determination in parabolic equations solved as moment problem

The problem is to find a(t) y w(x; t) such that wt = a(t) (wx)x+r(x; t) under the initial condition w(x; 0) =fi(x) and the boundary conditions w(0; t) = 0 ; wx(0; t) = wx(1; t)+alfa w(1; t) about a region D ={(x; t); 0 <x < 1; t >0}. In addition it must be fulfilled the integral of w (x, t) with respect to x is equal to E(t) where fi(x) , r(x; t) and E(t) are known functions and alfa is an arbitrary real number other than zero.The objective is to solve the problem as an application of the inverse moment problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. In addition, the method is illustrated with several examples.

Nontrivial minimal surfaces in a class of Finsler 3-spheres

In this paper, we study the rotationally invariant minimal surfaces in the Bao–Shen's spheres, which are a class of 3-spheres endowed with Randers metrics [Formula: see text] of constant flag curvature K = 1, where [Formula: see text] are Berger metrics, [Formula: see text] are one-forms and k > 1 is an arbitrary real number. We obtain a class of nontrivial minimal surfaces isometrically immersed in the Bao–Shen's spheres, which is the first class of nontrivial minimal surfaces with respect to the Busemann–Hausdorff measure in Finsler spheres. Moreover, we also obtain a new class of explicit minimal surfaces in the classical Berger spheres [Formula: see text], which was expected to get in [F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl.28(5) (2010) 593–607].

Technology Tips: Constructing the Square Root of a Complex Number

The author uses The Geometer's Sketchpad first to construct the square root of an arbitrary real number and then to construct the square root of a complex number.

COMPUTER ALGEBRA REEXAMINATION OF THE SCALED PARTICLE THEORY FOR HARD-SPHERE AND LENNARD-JONES FLUIDS

In the present work, an extension of the scaled particle theory (ESPT) for fluid using computer algebra is developed to obtain an equation of state (EOS), for Lennard-Jones fluid. A suitable functional form for surface tension S(r,d,∊) is assumed with intermolecular separation r as a variable, given below: [Formula: see text] where m is arbitrary real number, and d and ∊ are related to physical property such as average or suitable molecular diameter and the binding energy of the molecule respectively. It is found that, for hard sphere fluid ∊ = 0, the above assumption when introduced in scaled particle theory (SPT) frame and choosing arbitrary real number, m = 1/3, the corresponding EOS is in good agreement with the computer simulation of molecular dynamics (MD) result. Furthermore, for the value of m = -1 it gives a Percus–Yevick (pressure), and for the value of m = 1, it corresponds Percus–Yevick (compressibility) EOS.