A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function Ψ. We determine certain new double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function Ψ given in the paper. We present some corollaries as particular cases of the main results.

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On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals

Laplace transform is a powerful tool for solving differential and integrodifferential equations in engineering sciences. The use of Laplace transform for the solution of differential or integrodifferential equations sometimes leads to the solutions in the Laplace domain that cannot be inverted to the real domain by analytic methods. Therefore, we need numerical methods to invert the solution to the real domain. In this work, we construct numerical schemes based on Laplace transform for the solution of fractional-order Volterra integrodifferential equations in the sense of the Atangana-Baleanu Caputo derivative. We propose two numerical methods for approximating the solution of fractional-order linear and nonlinear Volterra integrodifferential equations. In our scheme, the inverse Laplace transform is approximated using a contour integration method and Stehfest method. Some numerical experiments are performed to check the accuracy and efficiency of the methods. The results obtained using these methods are compared.

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Numerical Approximation of Fractional-Order Volterra Integrodifferential Equation

Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. Many generalizations of convex functions exists in literature. The main aim of the article is to develop fractional integral inequalities for generalized strongly modified h-convex functions. Based on obtained fractional type integral inequalities we give some applications to the means. Our results are extension and generalization of many existing results.

On some fractional integral inequalities for generalized strongly modified $h$-convex functions

Core-satellite Graphs. Clustering, Assortativity and Spectral Properties

For a simple graph G with vertex set V (G) and edge set E(G), a labeling φ : V (G) ∪ E(G) −→ {1, 2, . . . , k} is called a vertex irregular total klabeling of G if for any two different vertices x and y, their weights wt(x) ∗ The work was supported by the Higher Education Commission Pakistan. 148 A. AHMAD, K.M. AWAN, I. JAVAID AND SLAMIN and wt(y) are distinct. The weight wt(x) of a vertex x in G is the sum of its label and the labels of all edges incident with the given vertex x. The total vertex irregularity strength of G, denoted by tvs(G), is the smallest positive integer k for which G has a vertex irregular total k-labeling. In this paper, we study the total vertex irregularity strength of flower, helm, generalized friendship and web graphs.

https://ajc.maths.uq.edu.au/pdf/51/ajc_v51_p147.pdf

Total vertex irregularity strength of wheel related graphs.

In this paper, Petrović’s inequality is generalized for h−convex functions on coordinates with the condition that h is supermultiplicative. In the case, when h is submultiplicative, Petrović’s inequality is generalized for h−concave functions. Also particular cases for P−function, Godunova-Levin functions, s−Godunova-Levin functions and s−convex functions has been discussed.

An extension of Petrović's inequality for $h-$convex ($h-$concave) functions in plane

In this paper, we obtain Ostrowski-type bounds for the weighted Čebyšev functional. Also we give bounds of weighted Čebyšev functional in the case of Steffesen’s generalization of Čebyšev inequality.

/pdf/steffensen-s-generalization-of-cebysev-inequality-4jpwgiqobz.pdf

Steffensen's generalization of Čebyšev inequality

Multivariate Hardy and Littlewood inequalities on time scales

The new outcomes of the present paper are 
 
 q
 
 -analogues (
 
 q
 
 stands for quantum calculus) of Hermite-Hadamard type inequality, Montgomery identity, and Ostrowski type inequalities for 
 
 s
 
 -convex mappings. Some new bounds of Ostrowski type functionals are obtained by using Hölder, Minkowski, and power mean inequalities via quantum calculus. Special cases of new results include existing results from the literature.

Khalid Mahmood Awan

Papers

Total vertex irregularity strength of wheel related graphs.

An extension of Petrović's inequality for \(h-\)convex (\(h-\)concave) functions in plane

Steffensen's generalization of Čebyšev inequality

Multivariate Hardy and Littlewood inequalities on time scales

Ostrowski Type Inequalities for s -Convex Functions via q -Integrals