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RADIO LABELING AND RADIO NUMBER FOR GENERALIZED CATERPILLAR GRAPHS
NAZEER, SAIMA,KHAN, M. SAQIB,KOUSAR, IMRANA,NAZEER, WAQAS The Korean Society for Computational and Applied M 2016 Journal of applied mathematics & informatics Vol.34 No.5
A Radio labeling of the graph G is a function g from the vertex set V (G) of G to ℤ<sup>+</sup> such that |g(u) - g(v)| ≥ diam(G) + 1 - d<sub>G</sub>(u, v), where diam(G) and d(u, v) are diameter and distance between u and v in graph G respectively. The radio number rn(G) of G is the smallest number k such that G has radio labeling with max{g(v) : v ∈ V(G)} = k. We investigate radio number for some families of generalized caterpillar graphs.
Waqas Nazeer,Mobeen Munir,Abdul Rauf Nizami,Samina Kausar,강신민 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this note we establish a new non-convex hybrid iteration algorithm corresponding to Khan iterative process [4] and prove strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Moreover, the main results are applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. The results presented in this article are interesting extensions of some current results.
NAZEER, WAQAS,MUNIR, MOBEEN,NIZAMI, ABDUL RAUF,KAUSAR, SAMINA,KANG, SHIN MIN The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.3
In this note we establish a new non-convex hybrid iteration algorithm corresponding to Khan iterative process [4] and prove strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Moreover, the main results are applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. The results presented in this article are interesting extensions of some current results.
RADIO AND RADIO ANTIPODAL LABELINGS FOR CIRCULANT GRAPHS G(4k + 2; {1, 2})
Nazeer, Saima,Kousar, Imrana,Nazeer, Waqas The Korean Society for Computational and Applied M 2015 Journal of applied mathematics & informatics Vol.33 No.1
A radio k-labeling f of a graph G is a function f from V (G) to $Z^+{\cup}\{0\}$ such that $d(x,y)+{\mid}f(x)-f(y){\mid}{\geq}k+1$ for every two distinct vertices x and y of G, where d(x, y) is the distance between any two vertices $x,y{\in}G$. The span of a radio k-labeling f is denoted by sp(f) and defined as max$\{{\mid}f(x)-f(y){\mid}:x,y{\in}V(G)\}$. The radio k-labeling is a radio labeling when k = diam(G). In other words, a radio labeling is an injective function $f:V(G){\rightarrow}Z^+{\cup}\{0\}$ such that $${\mid}f(x)=f(y){\mid}{\geq}diam(G)+1-d(x,y)$$ for any pair of vertices $x,y{\in}G$. The radio number of G denoted by rn(G), is the lowest span taken over all radio labelings of the graph. When k = diam(G) - 1, a radio k-labeling is called a radio antipodal labeling. An antipodal labeling for a graph G is a function $f:V(G){\rightarrow}\{0,1,2,{\ldots}\}$ such that $d(x,y)+{\mid}f(x)-f(y){\mid}{\geq}diam(G)$ holds for all $x,y{\in}G$. The radio antipodal number for G denoted by an(G), is the minimum span of an antipodal labeling admitted by G. In this paper, we investigate the exact value of the radio number and radio antipodal number for the circulant graphs G(4k + 2; {1, 2}).
Radio and Radio Antipodal Labelings For Circulant Graphs
Saima Nazeer,Imrana Kousar,Waqas Nazeer 한국전산응용수학회 2015 Journal of applied mathematics & informatics Vol.33 No.1
A radio $k$-labeling $f$ of a graph $G$ is a function $f$ from $V(G)$ to $Z^{+}\cup\{0\}$ such that$d(x,y)+|f(x)-f(y)|\geq k+1$ for every two distinct vertices $x$ and $y$ of $G$, where $d(x,y)$ is the distance between any two vertices $x, y\in G$. The span of a radio $k$-labeling $f$ is denoted by $sp(f)$ and defined as max$\{|f(x)-f(y)|: x,y\in V(G)\}$. The radio $k$-labeling is a radio labeling when$k=\text{diam}(G)$. In other words, a radio labeling is an injective function $f:V(G)\rightarrow Z^{+}\cup\{0\}$ such that$$|f(x)-f(y)|\geq \text{diam}(G)+1-d(x,y)$$ for any pair of vertices $x, y \in G$. The radio number of $G$denoted by rn$(G)$, is the lowest span taken over all radio labelings of the graph. When $k=\text{diam}(G)-1$, a radio $k$- labeling is called a radioantipodal labeling. An antipodal labeling for a graph $G$ is a function $f:V(G)\rightarrow\{0, 1, 2, ...\}$such that $d(x,y)+|f(x)-f(y)|\geq \text{diam}(G)$ holds for all $x,y\in G$. The radio antipodal number for $G$ denoted by an$(G)$, is the minimum span of an antipodal labeling admitted by $G$. In this paper, we investigate the exact value of the radio number and radio antipodal number for thecirculant graphs $G(4k+2;\{1,2\})$.
Radio labeling and Radio Number For Generalized Caterpillar Graphs
Saima Nazeer,M. Saqib Khan,Imrana Kausar,Waqas Nazeer 한국전산응용수학회 2016 Journal of applied mathematics & informatics Vol.34 No.5
A Radio labeling of the graph $G$ is a function $g$ from the vertex set $V(G)$ of $G$ to $\mathbb{Z}^{+}$ such that $|g(u)-g(v)|\geq\text{diam}(G)+1-d_G(u,v)$, where diam$(G)$ and $d(u,v)$ are diameter and distance between $u$ and $v$ in graph $G$ respectively. The radio number rn$(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling with max$\{g(v):v\in V(G)\}=k$. We investigate radio number for some families of generalized caterpillar graphs.
STRONG CONVERGENCE OF NEW VISCOSITY RULES OF NONEXPANSIVE MAPPINGS
AHMAD, MUHAMMAD SAEED,NAZEER, WAQAS,MUNIR, MOBEEN,NAQVI, SAYED FAKHAR ABBAS,KANG, SHIN MIN The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.5
The aim of this paper is to present two new viscosity rules for nonexpansive mappings in Hilbert spaces. Under some assumptions, the strong convergence theorems of the purposed new viscosity rules are proved. Some applications are also included.
STRONG CONVERGENCE OF NEW VISCOSITY RULES OF NONEXPANSIVE MAPPINGS
Muhammad Saeed Ahmad,Waqas Nazeer,Mobeen Munir,Sayed Fakhar Abbas Naqvi,강신민 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.5
The aim of this paper is to present two new viscosity rules for nonexpansive mappings in Hilbert spaces. Under some assumptions, the strong convergence theorems of the purposed new viscosity rules are proved. Some applications are also included.
Kang Shin Min,Farid Ghulam,Nazeer Waqas,Usman Muhammad 경남대학교 수학교육과 2019 Nonlinear Functional Analysis and Applications Vol.24 No.1
In this paper we have established a new identity for Katugampola fractional integrals. By using it we have found some generalizations of Riemann-Liouville fractional integral inequalities of Ostrowski type for (α,m)-convex functions. Also we prove some inequalities by taking particular appropriate values of α and m.