Triangular Snake Research Articles

3-Difference cordial labeling of some path related graphs

<p>Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map where k is an integer 2 ≤ k ≤ p. For each edge uv, assign the label |f(u) − f(v)|. f is called k-difference cordial labeling of G if |vf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x, 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 3-difference cordial labeling behavior of triangular snake, alternate triangular snake, alternate quadrilateral snake, irregular triangular snake, irregular quadrilateral snake, double triangular snake, double quadrilateral snake, double alternate triangular snake, and double alternate quadrilateral snake.</p>

Odd Vertex Equitable Even Labeling of Cycle Related Graphs

Let G be a graph with p vertices and q edges and A = {1, 3, ..., q} if q is odd or A = {1, 3, ..., q + 1} if q is even. A graph G is said to admit an odd vertex equitable even labeling if there exists a vertex labeling f : (G) → A that induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv such that for all a and b in A, |vf(a) − vf(b)| ≤ 1 and the induced edge labels are 2, 4, ..., 2q where vf(a) be the number of vertices v with f(v) = a for a ∈ A. A graph that admits an odd vertex equitable even labeling is called an odd vertex equitable even graph. Here, we prove that the subdivision of double triangular snake (S(D(Tn))), subdivision of double quadrilateral snake (S(D(Qn))), DA(Qm) ⊙ nK1 and DA(Tm) ⊙ nK1 are odd vertex equitable even graphs.

On b-coloring of central graph of some graphs

The b-chromatic number of G, denoted by ϕ(G), is the maximum k for which G has a b-coloring by k colors. A b-coloring of G by k colors is a proper k-coloring of the vertices of G such that in each color class i there exists a vertex x_{i} having neighbors in all the other k-1 color classes. Such a vertex x_{i} is called a b-dominating vertex, and the set of vertices {x₁,x₂…x_{k}} is called a b-dominating system. In this paper, we are going to investigate on the b-chromatic number of Central graph of Triangular Snake graph, Sunlet graph, Helm Graph, Double Triangular Snake graph, Gear graph, and Closed Helm graph are denoted as C(T_{n}), C(S_{n}), C(H_{n}), C(DT_{n}), C(G_{n}), C(CH_{n}) respectively.

On some labelings of triangular snake and central graph of triangular snake graph

A Triangular snake Tn is obtained from a path u1u2 …un by joining ui and ui+1 to a new vertex wi for 1≤i≤n−1. A Central graph of Triangular snake C(Tn) is obtained by subdividing each edge of Tn exactly once and joining all the non adjacent vertices of Tn. In this paper the ways to construct square sum, square difference, Root Mean square, strongly Multiplicative, Even Mean and Odd Mean labeling for Triangular Snake and Central graph of Triangular Snake graphs are reported.

Odd Harmonious Labeling of Some Classes of Cycle Related Graphs

A connected graph G = (V, E) of order atleast two, with order and size is called odd harmonious, if there exists an injection f:V → {0, 1, 2,, 2q − 1} such that the induced function f*:E → {, 3,, 2q1} defined by f*(uv) = [f (u) + f (v)], uv ∈ E is a bijection. Then f is called odd harmonious labeling of G. In this paper, it is proved that the Actinia graph, nonlinear symmetrical subdivisions of triangular snake, symmetrical subdivision of triangular snake, quadrilateral snake are odd harmonious.

Strongly Multiplicative Labeling of some Standard Graphs

In this paper we investigate strongly multiplicative labeling of some standard graphs. We prove that helm, flower graph, fan, friendship graph and bistar are strongly multiplicative. We also prove that gear graph, double triangular snake, double fan and double wheel are strongly multiplicative.

Cordialness of Arbitrary Supersubdivision of Graphs

In this paper we prove that arbitrary supersubdivision of ladder, cyclic ladder, triangular snake and certain double triangular snake are cordial.

Graceful and Odd Graceful Labeling of Graphs

In this paper we prove gracefulness and odd gracefulness of arbitrary super -subdivision of triangular snake, where labeling of the vertices follows arithmetic progression.

New results on vertex equitable labeling

The concept of vertex equitable labeling was introduced in [9]. A graph $G$ is said to be vertex equitable if there exists a vertex labeling $f$ such that for all $a$ and $b$ in $A$, $\left|v_f(a)-v_f(b)\right|\leq1$ and the induced edge labels are $1, 2, 3,\cdots, q$. A graph $G$ is said to be a vertex equitable if it admits a vertex equitable labeling. In this paper, we prove that the graphs, subdivision of double triangular snake $S(D(T_n))$, subdivision of double quadrilateral snake $S(D(Q_n))$, subdivision of double alternate triangular snake $S(DA(T_n))$, subdivision of double alternate quadrilateral snake $S(DA(Q_n))$, $DA(Q_m)\odot nK_1$ and $DA(T_m)\odot nK_1$ admit vertex equitable labeling.

ADCSS-Labeling of 2-Tuple Graphs of Some Graphs

A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. In this paper, we introduce the new concept, an absolute difference of cubic and square sum labeling of a graph. The graph for which every edge label is the absolute difference of the sum of the cubes of the end vertices and the sum of the squares of the end vertices. It is also observed that the weights of the edges are found to be multiples of 2. Here we characterize 2-tuple graphs of middle graph of paths and cycles, crown graph ,star graphs , the triangular snake graph, quadrilateral snake graph for adcss labeling.

Ladder and Subdivision of Ladder Graphs with Pendant Edges are Odd Graceful

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln + mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln) + mk1 is odd graceful. Finally, we prove that all the subdivision of triangular snakes (delt-k snake) with pendant edges S(delt-k snake)+ mk are odd graceful.

Square Difference 3-Equitable Labeling of Some Graphs

A square difference 3-equitable labeling of a graph G with vertex set V is a bijection f from V tof1;2;:::;j Vjg such that if each edge uv is assigned the label -1 ifj[f (u)] 2 [f (v)] 2 j 1(mod4), the label 0 ifj[f (u)] 2 [f (v)] 2 j 0(mod4) and the label 1 ifj[f (u)] 2 [f (v)] 2 j 1(mod4), then the number of edges labeled with i and the number of edges labeled with j differ by atmost 1 for 1 i;j 1. If a graph has a square difference 3-equitable labeling, then it is called square difference 3-equitable graph. In this paper, we investigate the square difference 3-equitable labeling behaviour of middle graph of paths, fan graphs, (P2n;S1), mK3, triangular snake graphs and friendship graphs.

Some Results on Super Root Square Mean Labeling

Let $G$ be a $(p,q)$ graph and $f:V(G)\leftarrow \{1,2,3,…,p+q \}$ be an injective function. For each edge $e=uv$ , let $f^*(e=uv)=\lceil \sqrt{\frac{f(u)^2+f(v)^2}{2}} \rceil$ or =\lfloor\sqrt{\frac{f(u)^2+f(v)^2}{2}} \rfloor$, then $f$ is called a super root square mean labeling if $f(V) \cup \{f^*(e):e \in E(G) \}=\{1,2,…,p+q \}. A graph that admits a super root square mean labeling is called as super root square mean graph. In this paper we prove that Double triangular snake, Alternate double triangular snake, Double quadrilateral snake and Alternate Double quadrilateral snake graphs are super root square mean graphs. Keywords: Root Square mean graph, Super Root Square mean graph, Triangular snake, Double triangular snake, Quadrilateral snake, Double quadrilateral snake.

Some Path Related 4-Cordial Graphs

In this paper, we discuss 4-cordial labeling of some path related graphs. We prove that middle graph, total graph and splitting graph of the path are 4-cordial. In addition to this we prove that $P_n^2 $ and triangular snakes are 4-cordial.

Some results on edge pair sum labeling

An injective map f:E(G)→{±1,±2,⋯,±q} is said to be an edge pair sum labeling if the induced vertex function f⁎:V(G)→Z−{0} defined by f⁎(v)=∑eϵEvf(e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f⁎(V(G)) is either of the form {±k1,±k2,⋯,±kp2} or {±k1,±k2,⋯,±kp−12}⋃{kp+12} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn⋃Cn and K1,n⋃K1,m are edge pair sum graphs.

$b$-Chromatic number of Some Splitting Graphs

A $b$-colouring of a graph $G$ is a proper vertex colouring of $G$ such that each colour class contains a vertex that has atleast one neighbour in every other colour class and $b$-chromatic number of a graph $G$ is the largest integer $\phi(G)$ for which $G$ has a $b$-colouring with $\phi(G)$ colours. In this paper, we have obtained the $b$-chromatic number of the splitting graphs of path $P_n$, cycle $C_n$, star $K_{1,n}$, fan graph $F_n$, triangular snake $T_n$, the $H$-graph $H_n$, the corona graph $P_n\circ K_1$ and $C_n\circ K_1$.

Some More Results on Root Square Mean Graphs

A graph with vertices and edges is called a Root Square Mean graph if it is possible to label the vertices with distinct elements from in such a way that when each edge is labeled with or , then the resulting edge labels are distinct. In this case is called a Root Square Mean labeling of . The concept of Root Square Mean labeling was introduced by (S. S. Sandhya, S. Somasundaram and S. Anusa). We investigated the Root Square Mean labeling of several standard graphs such as Path, Cycle, Comb, Ladder, Triangular snake, Quadrilateral snake etc., In this paper, we investigate the Root Square Mean labeling for Double Triangular snake, Alternate Double Triangular snake, Double Quadrilateral snake, Alternate Double Quadrilateral snake, and Polygonal chain.

Harmonious coloring of central graphs of certain snake graphs

A harmonious coloring of a simple graph is the proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number of G, denoted by χh (G), is the least number of colors in a harmonious coloring of G. The paper gives the structural properties and the estimates of harmonious chromatic number of central graphs of Triangular snake graph C [Tn], Double triangular snake graph C [D (Tn)] and Diamond snake graph C [Dn].

Trimagic Labeling in Digraphs

In this paper we introduce (1, 1) indegree vertex trimagic labeling and (1, 1) outdegree vertex trimagic labeling for digraphs. We prove that the directed paths nP2(n ≥ 3), nP4, Pm ⋃ Pn ⋃ Pr directed triangular snake TSn and the directed cycle Cn (n ≥ 3) admit both (1, 1) indegree vertex trimagic labeling and (1, 1) outdegree vertex trimagic labeling. We also prove that the full directed binary tree Tn(n > 3) admit (1, 1) indegree vertex trimagic labeling.

Product cordial labeling for alternate snake graphs

The product cordial labeling is a variant of cordial labeling. Here we investigate product cordial labelings for alternate triangular snake and alternate quadrilateral snake graphs.