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In this paper we discuss harmonic mean labeling behaviour of some cycle related graphs such as duplication, joint sum of the cycle and identification of cycle. Also we investigate harmonic mean labeling behaviour of alternate triangular snake A(T n ), alternate quadrilateral snake A(Q n ).
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Let G ( p , q ) be graph that consists of p = | V | vertices and q = | E | edges, where is the set of vertices and E is the set of edges of G . A graph G ( p , q ) is odd harmonious if there exist an injective function f : V → {0, 1, 2, …, 2 q − 1} that induced a bijective function f ∗ : E → {1, 3, 5, …, 2 q − 1} defined by f ∗ ( uv ) = f ( u ) + f...
A harmonious coloring is a proper vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. In this paper harmonious coloring of certain snake graphs are studied. Some structural properties of them are discussed. Also, their harmonious chromatic number was obtained .
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.
Let f : V (G) → {1, 2,..., |V (G)|} be a bijection, and let us denote S = f(u) + f(v) and D = |f(u) − f(v)| for every edge uv in E(G). Let f' be the induced edge labeling, induced by the vertex labeling f, defined as f' : E(G) → {0, 1} such that for any edge uv in E(G), f' (uv)=1 if gcd(S, D)=1, and f' (uv)=0 otherwise. Let ef' (0) and ef' (1) be t...
Citations
... Some of the harmonic mean graphs are investigated by S. Meena and M. Sivasakthi in [4]. The concept of Harmonic Mean labeling of graph was introduced by S. Somasundaram, R. Ponraj and S.S. Sandhya [5,6] and they investigated the existence of Harmonic mean labeling of several family of graphs such as this concept was then studied by several authors and studied their behavior in [7], [8], [9], and [10]. ...
A graph G with p vertices and q edges is called a harmonic mean(HM) labeling if it is possible to label the vertices x∈v with distinct labels ρ(x) from {1,2,⋯,q+1} in such a way that each edge e=ab is labeled with ρ(ab)=⌈(2ρ(a)ρ(b))/(ρ(a)+ρ(b))⌉ or ⌊(2ρ(a)ρ(b))/(ρ(a)+ρ(b))⌋ then the edge labels are distinct.In this case ρ is called Harmonic mean(HM) labeling of G. In this paper we introduce new graphs obtained from triangular snake graph TS_n such as TS_n∘K_1, and prove that they are Harmonic Mean labeling graphs.
... Definition: 1.1 [8] A Graph G = (V , E) with p points and q lines is called a Harmonic mean graph if it is possible to label the points ∈V with distinct labels f(v) from {1,2,…,q+1} in such a way that when each line e = uv is labeled with f(uv) = In this case f is called Harmonic mean labeling of G. ...
... S.Somasundaram, R.Ponraj and S.S.Sandhya were introduced the concept of harmonic mean labeling of graphs. They investigated the existence of harmonic mean labeling of several family of graphs such as path, comb, cycle C n , complete graph K n complete bipartite graph K 2,2 , triangular snake T n , quadrilateral snake Q n , alternate triangular snake A(T n ), alternate quadrilateral snake A(Q n ), crown C n K 1 ,C n K 2 ,C n K 3 , dragon, wheel in [8][9][10]. The harmonic mean labeling of step ladder S(T n , P n K 2 ,C n K 2 , flower graph, L n K 2 , triangular ladder, double triangular snake D(T n ), alternate double triangular snake A(DT n ) double quadrilateral snake D(Q n ), alternate double quadrilateral snake A(DQ n ), Q n K 1 , (C m K 1 ) ∪ C n , (C m K 1 ) ∪ P n , are investigated by C.Jayasekaran, C.David raj and S.S.Sandhya [11,12]. ...
... R. Ponraj and D. Ramya introduced Super mean labeling of graphs in [5]. S. Somasundaram and S.S. Sandhya introduced the concept of Harmonic mean labeling in [6] and studied their behavior in [7,8,9]. In this paper, we introduce the concept of kharmonic mean labeling and we investigate k-harmonic mean labeling of some graphs. ...
... R. Ponraj and D. Ramya introduced Super mean labeling of graphs in [5]. S. Somasundaram and S.S. Sandhya introduced the concept of Harmonic mean labeling in [6] and studied their behavior in [7,8,9]. ...
... Ponraj and Ramya introduced Super mean labeling of graphs in [5]. Somasundaram and Sandhya introduced the concept of Harmonic mean labeling in [6] and studied their behavior in [7,8,9]. Sandhya and David Raj introduced Super harmonic labeling in [9]. ...
... In this paper we prove that several snakes related graphs are sum perfect square. Definition 1.7 ( [6]). The double alternating quadrilateral snakes DA(Qn) are obtained from a path {v1, v2, . . . ...
... − 2n or 6i = 4 − 2n.=⇒ i = 1 − n or i = 4−2n6 , which contradicts with the choice of i, as i ∈ N.Assume if possible {f * (ei (1) ), 1 ≤ i ≤ n 2 } = {f * (e(3)i ), 1 ≤ i ≤ n 2 }, for some i. =⇒ n + 4i − 4 = n + 4i − 2 or n + 4i − 4 = 2 − n − 4i.=⇒ −4 = −2 or 8i = 6 − 2n. =⇒ 4 = 2 or i = 6−2n 8 , which contradicts with the choice of i, as i ∈ N. Injectivity for edge labels for subcase 2: ...
Let G = (V;E) be a simple (p; q) graph and f : V (G) ! f0; 1; 2; : : : ; p 1g be a bijection. We de�ne f� : E(G) ! N by
f�(uv) = (f(u))2 + (f(v))2 + 2f(u) � f(v), 8uv 2 E(G). If f� is injective, then f is called sum perfect square labeling. A
graph which admits sum perfect square labeling is called sum perfect square graph. In this paper we prove that several
snakes related graphs are sum perfect square.
... S. Somasundaram and R. Ponraj introduced mean labeling of graphs in [4]. S. Somasundaram and S.S Sandhya introduced the concept of harmonic mean labeling in [5] and studied their behavior in [7], [8] and [9]. C. David Raj, S.S Sandhya and C. Jayasekaran introduced the concept of an one modulo three harmonic mean graphs in [6]. ...
A graph G with p vertices and q edges is said to be an one modulo three harmonic mean graph if there is a function φ from the vertex set of G to {1, 3, 4, … ,3 − 2, 3 } with φ is one-one and φ induces a bijection φ * from the edge set of G to {1, 4, …, 3q-2}, where φ * (e = uv) = � 2 () () ()+ () � or � 2 () () ()+ () � and the function φ is called as an one modulo three harmonic mean labeling of G. In this paper, we investicate one modulo three harmonic mean labeling of some graphs.
... Among other alternatives, include means such as the GM, more suited for non-normal statistical distributions, and the HM, less influenced by the existence of higher values. Both GM and HM are used in different fields as a centrality parameter instead of the traditional AM (Rossman 1990;Limbrunner, Vogel, and Brown 2000;Spizman and Weinstein 2008;Sandhya, Somasundaram, and Ponraj 2012). This study investigates new ways to estimate the overall mean point density on airborne lidar surveys that maximize the central (normal) values and minimize the non-normal values. ...
The distribution of the discrete-return point density in airborne lidar flights obtained from an oscillating mirror laser scanner is analysed and alternative formulations to determine its value are presented. The point density in a lidar swath varies and can best be fitted with a potential function. This study confirms that calculating the overall point density with traditional statistical parameters yields biased results owing to the abnormally high densities of the swath boundaries. New formulas for calculating the representative mean are proposed: a weighted arithmetic mean (WAM) based on a potential function; geometric mean (GM); and harmonic mean (HM). All three means give more weight to the central sectors across the strip and less to the boundary sectors where extreme data redundancy exists. The WAM based on a potential function yields equivalent estimates as the HM; the GM yields slightly higher estimates. The results obtained improve the mean estimation and, more importantly, allow users to estimate better the mean point density on airborne lidar surveys, which are usually overestimated approximately by 15%.
... An alternate triangular snake A(T n ) is obtained from a path u 1 , u 2 , …,u n by joining u i and u i+1 (alternatively) to new vertex v i .That is every alternative edge of a path is replaced by a cycle C 3 .[7] An alternate quadrilateral snake A(Q n ) is obtained from a path u 1 , u 2 , …,u n by joining u i and u i+1 (alternatively) to new vertex v i ,w i respectively and then joining v i and w i .That is every alternative edge of a path is replaced by a cycle C 4 . ...
... S. Somasundaram and S.S.Sandhya introduced Harmonic mean labeling of graphs in [4] and studied their behaviour in [5] and [6]. In this paper, we investigate Harmonic mean labeling behaviour of Double Quadrilateral snake and Alternate Quadrilateral snake graphs. ...
In this paper we investigate Harmonic mean Labeling behaviour of Double Quadrilateral snake and Alternate Double Quadrilateral snake graphs.
