New Construction on SD-Prime Cordial Labeling

For every automorphism f of the free group F = (xl, ... ,, -) Gersten [2, Theorem 5.6] constructed a finite graph representing f and used this graph to give the first proof of Scott's conjecture that the subgroup of fixed points of f is finitely generated. A simpler construction of a graph representing f is contained in a more general treatment of Stallings [6]; also, as Goldstein and Turner [3] show, this graph can be obtained from Whitehead's 3-dimensional model for f . Apart from the applications in the study of fixed points, graphs representing automorphisms are of some independent interest; Hoare [4] uses them to reprove Whitehead's Cut Vertex lemma [7] and Thurston's Bounded Cancellation theorem [1]. This note offers another construction of Gersten's graph with a very short proof. We say that the graph F represents f if F has the following properties. It is finite with every edge colored blue or red and labeled xi or x-1 ( 1 < i < n), assuming that inverse edges have the same color and inverse labels. Every xl is the label of at least one blue and one red edge. Equally colored edges emanating from the same vertex have different labels. The set of all blue edges, and the set of red edges as well, is the edge set of a maximal tree in F. Finally, there is a vertex * such that for every path p which begins and ends at * one has blue (p) f(red(p)), where blue (p) is the element of F (a word in x obtained by reading out the labels of blue edges while traversing p, and red(p) is defined analogously. (This is a modified definition, an inessential variation of the original one.) Let T be the Cayley graph of F [5, p. 123] and TB and TR copies of T colored blue and red respectively. Let FO be the graph obtained by taking TB and TR together and identifying everyvertex v of TB with the vertex fv -l of TR. It is readily seen that Fo satisfies all conditions of the preceding paragraph except that of being finite. If elements of F are regarded as reduced words in x1 , it is easy to see that a subset A of F containing 1 is the vertex set of a subtree of T if and

A k-Zumkeller labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that each edge uv ∈ E is assigned the label f (u) f (v), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we prove that the graph Pm × Pn is k-Zumkeller graph for m, n ≥ 3 while Pm × Cn and Cm × Cn are k-Zumkeller graphs for n ≡ 4 (mod2). Also we show that the graphs Pm ⊗ Pn and Pm ⊗ Cn for m, n ≥ 3 admit k-Zumkeller labeling. Further, the graph Cm ⊗ Cn where m or n is even admit a k-Zumkeller labeling.

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, &lt;K1,n:n&gt; are Extended Mean Cordial Graphs.

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

We describe a new random greedy algorithm for generating regular graphs of high girth: Let k ≥ 3 and c ∈ (0, 1) be fixed. Let ℕ be even and set . Begin with a Hamilton cycle G on n vertices. As long as the smallest degree , choose, uniformly at random, two vertices u, v ∈ V(G) of degree whose distance is at least g − 1. If there are no such vertex pairs, abort. Otherwise, add the edge uv to E(G). We show that with high probability this algorithm yields a k‐regular graph with girth at least g. Our analysis also implies that there are labeled k‐regular n‐vertex graphs with girth at least g.

A bijection f: V(G) → {1,2,··· ,|V(G)|} induces an edge labeling f∗: E(G) → {0,1} such that for any edge uv in G, f∗(uv) = 1 if gcd(S,D) = 1 and f∗(uv) = 0 otherwise, where S = f(u)+ f(v) and D = |f(u)− f(v)|. The labeling f is called SD-prime cordial labeling if |ef∗(0) − ef∗(1)|≤1. We say that G is SD-prime cordial graph if it admits SD-prime cordial labeling. In this paper, we prove that certain classes of zero-divisor graphs of commutative rings are SD-prime cordial graphs.

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.

Let G be a graph of order p. A numbering f of G is a labeling that assigns distinct elements of the set 1,2,…,p to the vertices of G, where each edge uv of G is labeled fu+fv. The strength strfG of a numbering f:VG→1,2,…,p of G is defined by strfG=maxfu+fvuv∈EG,that is, strfG is the maximum edge label of G, and the strength strG of a graph G itself is strG=minstrfGf is a numbering of G.The strengths strT and strTn,k are determined for caterpillars T and k-level complete n-ary trees Tn,k. The strength strG is also given for graphs G obtained by taking the corona of certain graphs and an arbitrary number of isolated vertices. The work of this paper suggests an open problem on the strength of trees.

(G);E(G)) with p vertices and q edges, if there exists an injectionfunction : V (G) ! f0; 1; 2; ; 2q ô€€€ 1g with each edge uv assignedthe label jf(u)ô€€€f(v)j, the resulting edge labels are f1; 3; 5; ; 2qô€€€1g.The tensor product of two graphs, G and H, has a vertex setV (G) V (H) and an edge between (u; v) and (u0; v0), i both uu0 2 E(G) andvv0 2 E(H), here we denote tensor product by ^. In this paper, weprove Sn ^ Sm, Sn ^ Pm and Cn ^ Sm are odd graceful.

A k-edge product cordial labelling is a variant of the well-known cordial labelling. In this paper, a balanced k-edge product cordial labelling is suggested and some sufficient conditions for balanced 3-edge product cordial graphs are proved. Moreover, a construction of graphs admitting a balanced 3-edge product cordial labelling is presented.

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

Cordial labeling of hypertrees

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

In this paper, we investigate some new families of edge sum divisor cordial graphs, face edge sum divisor cordial graphs and total face edge sum divisor cordial graphs.Keywords -Edge sum divisor cordial labelling, Edge sum divisor cordial graph, face edge sum divisor cordial graph, total face edge sum divisor cordial graph, switching of any vertex.

An injective function f : V (G) → {T0, T1, T2, . . . , Tn}, where n = |V (G)| − 1, is said to be a Tribonacci cordial labeling if the induced function f ∗ E(G) →{0, 1} defined by f∗(uv) = (f(u) + f(v)) mod 2 satisfies the condition |ef (0) − ef (1)| ≤ 1 where ef (0) is the number of edges with label 0 and ef (1) is the number of edges with label 1. A graph that admits a tribonacci cordial labeling is called a Tribonacci cordial graph. In this paper, we determined the Tribonacci Cordial Labeling of Triangular Snake Graph T Sn, Double Triangular Snake Graph D(T Sn), Quadrilateral Snake Graph QSn, Double Quadrilateral Snake Graph D(QSn), and Cycle Quadrilateral Snake Graph C(QSn).