r. ponraj

In this paper we investigate the pair difference cordial labelingbehaviour of diamond ladder graph,lattitude ladder, octopus graph,pagodagraph, planter graph and semi jahangir graph. Prime labeling behaviour of plantergraph, duplication of planter graph, fusion of planter graph,switching of plantergraph, joining of two copies of planter graph were studied by A.Edward samueland S.Kalaivani[4]. Dafik, Riniatul Nur Wahidah hve been examined the RainbowAntimagic coloring of special graphs like volcano, sandat graph, sunflower,octpus,semijahangir [3]. Prihandini,R M., at.el have been studied the elegant labeling ofshackle graphs and diamond ladder graphs [16]. Classical meanness of some graphssuch as one-side step graph,double-sided step graph,grid,slanding ladder,diamondladder,lattitude ladder was studied by Alanazi et. al [1]. In [17] Yeni Susanti et.al studied the edge odd geaceful labeling behaviour of prism, antiprism, cartesianproduct graphs. The notion of pair diference cordial labeling of a graph was introduced in [7].

Let G be a graph. Let f : V (G) → {0, 1, 2,... ,k − 1}be a function where k ∈ N and k > 1. For each edge uv, assign thelabel f (uv) = lf(u)+f(v)2m. f is called a k-total mean cordial labeling of G if |tmf (i) − tmf (j)| ≤ 1, for all i,j ∈ {0, 1, 2,... ,k − 1},where tmf (x) denotes the total number of vertices and edges labelled with x, x ∈ {0, 1, 2,... ,k − 1}. A graph with admit a k-totalmean cordial labeling is called k-total mean cordial graph. In thispaper we examine the 4-Total mean cordial labeling of some trees

Let $G$ be a graph. Let $f:V\left(G\right)\rightarrow \left\{0,1,2,\ldots,k-1\right\}$ be a function where $k\in \mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $f\left(uv\right)=\left\lceil \frac{f\left(u\right)+f\left(v\right)}{2}\right\rceil$. $f$ is called a $k$-total mean cordial labeling of $G$ if $\left|t_{mf}\left(i\right)-t_{mf}\left(j\right) \right| \leq 1$, for all $i,j\in\left\{0,1,2,\ldots,k-1\right\}$, where $t_{mf}\left(x\right)$ denotes the total number of vertices and edges labelled with $x$, $x\in\left\{0,1,2,\ldots,k-1\right\}$. A graph with admit a $k$-total mean cordial labeling is called $k$-total mean cordial graph. In this paper we investigate the $4$-total mean cordial labeling behaviour of some spider graph.

In this paper we investigate the pair difference cordial labeling behavior of double alternate triangular snake and double alternate quadrilatral snake graphs.

In this paper, we investigate the pair difference cordial labeling behaviour of some star related graphs.

In this paper, we consider only finite, undirected, and simple graphs. The concept of cordial labeling was introduced by Cahit[4]. Different types of cordial-related labeling were studied in [1, 2, 3, 5, 16]. In a similar line, the notion of pair difference cordial labeling of a graph was introduced in [8]. The pair difference cordial labeling behavior of several graphs like path, cycle, star, wheel, triangular snake, alternate triangular snake, butterfly, ladder, Mobius ladder, slanting ladder, and union of some graphs have been investigated in [ 8, 9, 10, 11, 12, 13, 14, 15]. The $m-$ copies of a graph $G$ is denoted by $mG$ [7]. In this paper, we investigate the pair difference cordial labeling behavior of $m-$ copies of Path, Star, Cycle, and Ladder graphs.

Let $G$ be a graph. Let $f:Vleft(Gright)rightarrow left{0,1,2,ldots,k-1right}$ be a function where $kin mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $fleft(uvright)=leftlceil frac{fleft(uright)+fleft(vright)}{2}rightrceil$. $f$ is called $k$-total mean cordial labeling of $G$ if $left|t_{mf}left(iright)-t_{mf}left(jright) right| leq 1$, for all $i,jinleft{0,1,2,ldots,k-1right}$, where $t_{mf}left(xright)$ denotes the total number of vertices and edges labelled with $x$, $xinleft{0,1,2,ldots,k-1right}$. A graph with admit a $k$-total mean cordial labeling is called $k$-total mean cordial graph. In this paper, we investigate the $4$-total mean cordial labeling of some graphs derived from the complete bipartite graph $K_{2,n}$.

In this paper, we introduce a new graph labeling called pair mean cordial labeling of graphs. Also, we investigate the pair mean cordiality of some graphs like path, cycle, complete graph, star, wheel, ladder, and comb.

Let $G = (V, E)$ be a $(p,q)$ graph.Define begin{equation*}rho =begin{cases}frac{p}{2} ,& text{if $p$ is even}\frac{p-1}{2} ,& text{if $p$ is odd}\end{cases}end{equation*}\ and $L = {pm1 ,pm2, pm3 , cdots ,pmrho}$ called the set of labels.noindent Consider a mapping $f : V longrightarrow L$ by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when $p$ is odd.The labeling as defined  above is said to be a pair difference cordial labeling if for each edge $uv$ of $G$ there exists a labeling $left|f(u) - f(v)right|$ such that $left|Delta_{f_1} - Delta_{f_1^c}right| leq 1$,  where $Delta_{f_1}$ and $Delta_{f_1^c}$ respectively denote the number of edges labeled with $1$ and number of edges not labeled with $1$. A graph $G$ for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behavior of planar grid and mangolian tent graphs.

*The formulas are not displayed.

noindent Let $G = (V, E)$ be a $(p,q)$ graph.\Define begin{equation*}rho =begin{cases}frac{p}{2} ,& text{if $p$ is even}\frac{p-1}{2} ,& text{if $p$ is odd}\end{cases}end{equation*}\ and $L = {pm1 ,pm2, pm3 , cdots ,pmrho}$ called the set of labels.\noindent Consider a mapping $f : V longrightarrow L$ by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when $p$ is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge $uv$ of $G$ there exists a labeling $left|f(u) - f(v)right|$ such that $left|Delta_{f_1} - Delta_{f_1^c}right| leq 1$, where $Delta_{f_1}$ and $Delta_{f_1^c}$ respectively denote the number of edges labeled with $1$ and number of edges not labeled with $1$. A graph $G$ for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of some snake and butterfly graphs.

Let $G$ be a graph. Let $f:Vleft(Gright)rightarrow left{0,1,2,ldots,k-1right}$ be a function where $kin mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $fleft(uvright)=leftlceil frac{fleft(uright)+fleft(vright)}{2}rightrceil$. $f$ is called $k$-total mean cordial labeling of $G$ if $left|t_{mf}left(iright)-t_{mf}left(jright) right| leq 1$, for all $i,jinleft{0, 1, ldots, k-1right}$, where $t_{mf}left(xright)$ denotes the total number of vertices and edges labelled with $x$, $xinleft{0,1,2,ldots,k-1right}$. A graph with admit a $k$-total mean cordial labeling is called $k$-total mean cordial graph.

In this paper, an alternative proof is provided for a theorem of R.L.Graham concerning Chebyshev polynomials.  While studying the properties of a double star, R.L.Graham [2] proved a theorem concerning Chebyshev polynomials of the first kind ${T_n (x)}$. The purpose of this paper is to provide an alternative proof for his theorem. Our method is based on the divisibility properties of the natural numbers. One may observe that the Chebyshev polynomials evaluated at integers considered by R.L.Graham match with the solutions of the Pell's equation for a general, square-free $D in N$.

Let $G$ be a graph. Let $f:Vleft(Gright)rightarrow left{0,1,2,ldots,k-1right}$ be a function where $kin mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $fleft(uvright)=leftlceil frac{fleft(uright)+fleft(vright)}{2}rightrceil$.  $f$ is called $k$-total mean cordial labeling of $G$ if $left|t_{mf}left(iright)-t_{mf}left(jright) right| leq 1$, for all $i,jinleft{0,1,2,ldots,k-1right}$, where $t_{mf}left(xright)$ denotes the total number of vertices and edges labelled with $x$, $xinleft{0,1,2,ldots,k-1right}$.  A graph with admit a $k$-total mean cordial labeling is called $k$-total mean cordial graph.

vspace{0.2cm} Let $G$ be a graph and $f:V(G)rightarrow {1,2,3,.....left|V(G)right|}$ be a bijection. Let $p_{uv}=f(u)f(v)$ and\ $ d_{uv}= begin{cases} left[frac{f(u)}{f(v)}right] ~~if~~ f(u) geq f(v)\ \ left[frac{f(v)}{f(u)}right] ~~if~~ f(v) geq f(u)\ end{cases} $\ for all edge $uv in E(G)$. For each edge $uv$ assign the label $1$ if $gcd (p_{uv}, d_{uv})=1$ or $0$ otherwise. $f$ is called PD-prime cordial labeling if $left|e_{f}left(0right)-e_{f}left(1right) right| leq 1$ where $e_{f}left(0right)$ and $e_{f}left(1right)$ respectively denote the number of edges labelled with $0$ and $1$. A graph with admit a PD-prime cordial labeling is called PD-prime cordial graph. & & vspace{0.2cm}.The formula is not displayed correctly!

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a $k$-total difference cordial labeling is called a $k$-total difference cordial graphs. We investigate $k$-total difference cordial labeling of some graphs and study the $3$-total difference cordial labeling behaviour of star,bistar,complete bipartiate graph,comb,wheel,helm,armed crown etc.the formula is not displayed correctly!

Let $G$ be a $(p,q)$ graph. Let $f:V(G)to{1,2, ldots, k}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $gcd(f(u),f(v))$. $f$ is called $k$-Total prime cordial labeling of $G$ if $left|t_{f}(i)-t_{f}(j)right|leq 1$, $i,j in {1,2, cdots,k}$ where $t_{f}(x)$ denotes the total number of vertices and the edges labelled with $x$. A graph with a $k$-total prime cordial labeling is called $k$-total prime cordial graph. In this paper we investigate the $4$-total prime cordial labeling of some graphs like Prism, Helm, Dumbbell graph, Sun flower graph.the formula is not displayed correctly!

In this paper we introduce a new graph labeling method called k-Total prime cordial. Let G be a (p,q) graph. Let f:V(G)to{1,2, ldots, k} be a map where k in mathbb{N} and k>1. For each edge uv, assign the label gcd(f(u),f(v)). f is called k-Total prime cordial labeling of G if left|t_{f}(i)-t_{f}(j)right|leq 1, i,j in {1,2, ldots, k} where t_{f}(x) denotes the total number of vertices and the edges labeled with x. We investigate k-total prime cordial labeling of some graphs and study the 4-total prime cordial labeling of path, cycle, complete graph etc.the formula is not displayed correctly!

Let G be a (p,q) graph and A be a group. We denote the order of an element a in A by o(a).  Let f:V(G)rightarrow A be a function. For each edge uv assign the label 1 if (o(f(u)),o(f(v)))=1 or 0 otherwise. f is called a group A Cordial labeling if |v_f(a)-v_f(b)| leq 1 and |e_f(0)- e_f(1)|leq 1, where v_f(x) and e_f(n) respectively denote the number of vertices labelled with an element x and number of edges labelled with n (n=0,1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1 ,-1 ,i ,-i} Cordial graphs and characterize the graphs C_n + K_m (2 leq m leq 5) that  are group {1 ,-1 ,i ,-i} Cordial.the formula is not displayed correctly!

In this paper we generalize the remainder cordial labeling, called $k$-remainder cordial labeling and investigate the $4$-remainder cordial labeling behavior of certain graphs.

In this paper we introduce remainder cordial labeling of graphs. Let G be a (p,q) graph. Let f:V(G)rightarrow {1,2,...,p} be a 1-1 map. For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) or f(v) is divided by f(u) according as f(u)geq f(v) or f(v)geq f(u). The functionf is called a remainder cordial labeling of G if left| e_{f}(0) - e_f(1) right|leq 1 where e_{f}(0) and e_{f}(1) respectively denote the number of edges labelled with even integers and odd integers. A graph G with a remainder cordial labeling is called a remainder cordial graph. We investigate the remainder cordial behavior of path, cycle, star, bistar, crown, comb, wheel, complete bipartite K_{2,n} graph. Finally we propose a conjecture on complete graph K_{n}.the formula is not displayed correctly!

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f (u), f(v)). f is called k-prime cordial labeling of G if |vf</sub> (i) − vf</sub> (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef</sub> (0) − ef</sub> (1)| ≤ 1 where vf</sub> (x) denotes the number of vertices labeled with x, ef</sub> (1) and ef</sub> (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 3- prime cordial labeling behavior of union of a 3-prime cordial graph and a path Pn</sub>.the formula is not displayed correctly!

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf</sub> (i) − vf</sub> (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef</sub> (0) − ef</sub> (1)| ≤ 1 where vf</sub> (x) denotes the number of vertices labeled with x, ef</sub> (1) and ef</sub> (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 4- prime cordial labeling behavior of complete graph, book, flower, mCn</sub> and some more graphs.the formula is not displayed correctly!

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.the formula is not displayed correctly!

A graph G = (V,E) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : V (G) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ V (G), xy ∈ E(G), and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). In this paper, we investigate the total mean cordial labeling of Cn2, ladder Ln, book Bm and some more graphs.the formula is not displayed correctly!