Totally vertex-magic cordial labeling

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.

On 3-total edge product cordial labeling of a carbon nanotube network

Let f be a function from V (G) to {0,1,2}. Give a numerical value (label) from {0, 1, 2} for every edge uv. f is called a mean cordial labeling if |v_f (i)-v_f (j)|≤1 and |e_f (i)-e_f (j)|≤1, ∀ i,j ∈ {0,1,2} where v_f (x) and e_f (x) represent the number of vertices and edges respectively labeled as x(x = 0,1,2). A graph with mean cordial labeling is called a mean cordial graph. In this paper, we study the mean cordial labeling pattern of shadow-graphs of path graphs D_2 (P_n ), for n≥2

Let G = (V,E) be a graph with p vertices and q edges. A Extended Mean Cordial Labeling of a Graph G with vertex set V is a bijection from V to {0, 1,2} such that each edge uv is assigned the label where ⌈ x ⌉ is the least integer greater than or equal to x with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by almost 1. The graph that admits an Extended Mean Cordial Labeling is called Extended Mean Cordial Graph. In this paper, we proved that tree related graphs Hdn, K 1,n, Tgn, <K1,n:n> are Extended Mean Cordial Graphs.

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let [Formula: see text] be a map from [Formula: see text] to [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. For each edge [Formula: see text] assign the label [Formula: see text] [Formula: see text]. [Formula: see text] is called a k-product cordial labeling if [Formula: see text], and [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] denote the number of vertices and edges, respectively, labeled with [Formula: see text] [Formula: see text]. A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling.

A mapping f:V(G)→{0,1,2} is called a 3-product cordial labeling if |vf(i)−vf(j)|≤1 and |ef(i)−ef(j)|≤1 for any i,j∈{0,1,2}, where vf(i) denotes the number of vertices labeled with i,ef(i) denotes the number of edges xy with f(x)f(y)≡i(mod 3). A graph with a 3-product cordial labeling is called a 3-product cordial graph. In this paper, we investigate the 3-product cordial behaviour for some standard graphs.

A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |vf(i) − vf(j)| ≤ 1 and |ef(i) − ef(j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where vf(i) denotes the number of vertices labeled with i, ef(i) denotes the number of edges xy with f(x)f(y) ≡ i(mod 3). A graph with 3-product cordial labeing is called 3-product cordial graph. In this paper we establish that switching of an apex vertex in closed helm, double fan, book graph K1,n × K2 and permutation graph P (K2 + mK1, I) are 3-product cordial graphs.