Triangular Snake Research Articles

An Application on Harmonic Mean Labeling of Variations in Triangular Snake Graphs

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.

Some New Results on Domination and Independent Dominating Set of Some Graphs

One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set <I>S</I>⊆<I>V</I> is said to be dominating set of <I>G</I> if for every <i>v </i>є <I>V</I>-<I>S</I> there exists a vertex <i>u</i> є <I>S</I> such that <i>uv</i> є <I>E</I>. The dominance number of <I>G</I>, represented by <i>γ</i>(<I>G</I>), is the lowest cardinality of vertices among the dominating set of <I>G</I>. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph <I>G</I> and input <I>K</I>, <i>γ</i>(<I>G</I>) ≤ <I>K</I>. This is known as the dominating set issue. Consequently, it is thought that calculating <i>γ</i>(<I>G</I>) for each given graph <I>G</I> may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset <I>S</I>, then <I>S</I>⊆<I>V</I> is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of <I>G</I> is a set <I>S</I> of vertices in a graph <I>G</I> that is both an independent and a dominating set of <I>G</I>. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.

Antimagic Labeling for Some Snake Graphs

A graph with q edges is called antimagic if its edges can be labeled with 1, 2, 3, ..., q without repetition such that the sums of the labels of the edges incident to each vertex are distinct. In this paper we study antimagic labeling of double triangular snake, alternate triangular snake, double alternate triangular snake, quadrilateral snake, double quadrilateral snake, alternate quadrilateral snake, double alternate quadrilateral snake.

Crossing Numbers of the Cartesian Product of the Double Triangular Snake Graphs With Path Pm.

The crossing number Cr(G) of a graph G is the least number of edge crossings in all possible good drawings of G in the plane. Join and Cartesian products of graphs have many interesting graph-theoretical properties. In this paper, we evaluate the crossing number of the Cartesian product of double triangular snake graph DT2 with the path Pm. In this paper, we proved Cr(DT2 × Pm) = 6(m − 2), form ≥ 2 where Cr denotes the crossing number

EDGE IRREGULAR REFLEXIVE LABELING ON ALTERNATE TRIANGULAR SNAKE AND DOUBLE ALTERNATE QUADRILATERAL SNAKE

Let G in this paper be a connected and simple graph with set V(G) which is called a vertex and E(G) which is called an edge. The edge irregular reflexive k-labeling f on G consist of integers {1,2,3,...,k_e} as edge labels and even integers {0,2,4,...,2k_v} as the label of vertices, k=max{k_e,2k_v}, all edge weights are different. The weight of an edge xy in G represented by wt(xy) is defined as wt(xy)= f (x)+ f (xy)+ f (y). The smallest k of graph G has an edge irregular reflexive k-labeling is called the reflexive edge strength, symbolized by res (G). In article, we discuss about edge irregular reflexive k-labeling of alternate triangular snake A(T_n ) and the double alternate quadrilateral snake DA(Q_n ). In this paper, the res of alternate triangular snake A(T_n ) , n≥3 has been obtained. That is ⌈(2n-1)/3⌉ for n even,2n-1≢2,3 (mod 6),⌈(2n-1)/3⌉+1 for n even,2n-1=2,3 (mod 6),⌈(2n-2)/3⌉ for n odd,2n-2≢2,3 (mod 6), and ⌈(2n-2)/3⌉+1 for n odd,2n-2=2,3 (mod 6). Then, the reflexive edge strength of double alternate quadrilateral snake DA (Q_n) ⌈ (4n-1 )/3⌉for n even, 4n - 1 ≠2,3 (mod 6), ⌈ (4n-1 )/3⌉+1 for n even, 4n - 1 = 2,3 (mod 6), ⌈ (4n-4)/3⌉ for n odd, 4n - 4 ≠2,3 (mod 6), and ⌈ (4n-4 )/3⌉+1 for n odd, 4n - 4 = 2,3 (mod 6).

A Study on Sum Divisor Cordial Labeling Graphs

Cordial labeling refers to a graph with vertices and edges as a sum divisor if there exists a bijection function such that for each edge assign the label 1 , if and 0 otherwise, satisfying the condition is the number of edges having the label 1 , and is the number of edges having the label 0 . A graph with sum-divisor cordial labeling is called a sum-divisor cordial graph. In the paper, we establish this alternate triangular belt graph, twig graph, duplication of the top vertex Alternate triangular snake graph, duplication of top vertex Pentagon snake graph, jellyfish when and are even, Spider graph with n spokes .

On Some Topological Indices of the Triangular Snake Graph and Alternate Triangular Snake Graph

The quantitative specifications which are used to narrate the atomic topology of graphs are usually entitled as topological indices in theoretical Chemistry. The features of physical and chemical properties like melting point, entropy, boiling point, energy generation and enthalpy of vaporization of chemical compounds can be estimated by means of these topological indices. The field of Graph theory has a remarkable application of linking certain graphs with many types of topological indices. In this paper we compute first and second Zagreb indices, Randi´c index, sum-connectivity index, harmonic index, inverse sum indeg index, modified first and second Zagreb indices and first and second hyper Zagreb indices of Triangular snake graph and alternate Triangular snake graph.

DIVISOR CORDIAL LABELING FOR SOME SNAKES AND DEGREE SPLITTING RELATED GRAPHS

For a graph G = (V (G),E(G)), the vertex labeling function is defined as a bijection f : V (G) → {1, 2, . . . , |V (G)|} such that an edge uv is assigned the label 1 if one f(u) or f(v) divides the other and 0 otherwise. f is called divisor cordial labeling of graph G if the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In 2011, Varatharajan et al. [24] have introduced divisor cordial labeling as a variant of cordial labeling. In this paper, we study divisor cordial labeling for triangular snake and quadrilateral snake. Moreover, we investigate divisor cordial labeling for the degree splitting graph of path, shell, cycle with one chord, crown and comb graph.

New Results on Face Magic Mean Labeling of Graphs

In the midst of the 1960s, a theory by Kotzig-Ringel and a study by Rosa sparked curiosity in graph labeling. Our primary objective is to examine some types of graphs which admit Face Magic Mean Labeling (FMML). A bijection <img src=image/13429259_01.gif> is called a (1,0,0) F-Face magic mean labeling [FMML] of <img src=image/13429259_02.gif> if the induced face labeling <img src=image/13429259_03.gif> <img src=image/13429259_04.gif> A bijection <img src=image/13429259_05.gif> is called a (1,1,0) F-Face magic mean labeling [FMML] of <img src=image/13429259_02.gif> if the induced face labeling <img src=image/13429259_06.gif> <img src=image/13429259_07.gif> In this paper it is being investigated that the (1, 0, 0) – Face Magic Mean Labeling (F-FMML) of Ladder graphs, Tortoise graph and Middle graph of a path graph. Also (1,0,0) and (1,1,0) F-Face Magic Mean Labeling is verified for Ortho Chain Square Cactus graph, Para Chain Square Cactus graph and some snake related graphs like Triangular snake graphs and Quadrilateral snake graphs. For a wide range of applications, including the creation of good kind of codes, synch-set codes, missile guidance codes and convolutional codes with optimal auto correlation characteristics, labeled graphs serve as valuable mathematical models. They aid in the ability to develop the most efficient non-standard integer encodings; labeled graphs have also been used to identify ambiguities in the access protocol of communication networks; data base management to identify the best circuit layouts, etc.

Study of the Crossing Number Associated with Strong Product of Path with Cycle and Triangular Snake Graph

Study of the Crossing Number Associated with Strong Product of Path with Cycle and Triangular Snake Graph

Radio Labeling of Double Triangular Snake Graphs

Radio Labeling of Double Triangular Snake Graphs

Some Path Related Cordial Graphs

In this research paper, we prove that different families of snake graphs such as triangular snake graph with pendant edges, alternate triangular snake graph with and without pendant edges, quadrilateral snake graph with and without pendant edges, alternate quadrilateral snake graph with pendant edges and double quadrilateral snake graph with and without pendant edges are cordial graphs.

The Rainbow-Vertex Connection Number [RVCN] of Subdivision of Certain Graphs

Rainbow-Vertex Connection Number [rvcn] is computed for some graphs by the researchers. Here we have considered the subdivision graphs of certain graph classes. The rainbow edge connection number of subdivision of Triangular snake graph was already found[1]. Using the definition of rainbow-vertex connection number [5], which is the smallest positive integer k such that the graph is rainbow-vertex connected, we find the rainbow vertex number of subdivision graph of Friendship graph ,, Triangular snake graph and Comb graph .

Double Duplication of Special Classes of Cycle Related Graphs

Let <img src=image/13427481_01.gif> be a simple, finite, connected, plane graph with the vertex set <img src=image/13427481_02.gif>, the edge set <img src=image/13427481_03.gif> and the face set <img src=image/13427481_04.gif>). Martin Baca [1] defined, a connected plane graph <img src=image/13427481_01.gif> with vertex set <img src=image/13427481_02.gif>, edge set <img src=image/13427481_03.gif> and face set <img src=image/13427481_04.gif> to be <img src=image/13427481_05.gif> face antimagic if there exists positive integers <img src=image/13427481_06.gif> and <img src=image/13427481_07.gif> and a bijection <img src=image/13427481_08.gif>: <img src=image/13427481_09.gif> such that the induced mapping <img src=image/13427481_10.gif>: <img src=image/13427481_11.gif>, where for a face <img src=image/13427481_12.gif>, <img src=image/13427481_13.gif> is the sum of all <img src=image/13427481_14.gif> for all edges <img src=image/13427481_15.gif> surrounding <img src=image/13427481_12.gif> is also a bijection. This paper proves the existence of face antimagic labeling for the double duplication of all vertices by edges of gear graph <img src=image/13427481_16.gif> for <img src=image/13427481_17.gif>, grid graph <img src=image/13427481_18.gif> for <img src=image/13427481_19.gif>, where <img src=image/13427481_20.gif> even, prism graph <img src=image/13427481_21.gif> for <img src=image/13427481_22.gif> and the double duplication of all vertices by edges of strong face of triangular snake graph <img src=image/13427481_23.gif> for <img src=image/13427481_17.gif>. The <img src=image/13427481_05.gif> face antimagic labeling for double duplication of special graphs can be used to encrypt and decrypt the messages, which is used as a real time application. In [3], we used <img src=image/13427481_05.gif> face antimagic labeling of strong face of duplication of all vertices by edges of a tree <img src=image/13427481_23.gif> for <img src=image/13427481_24.gif> to encrypt and decrypt thirteen secret numbers which can be extended to double duplication of graphs to encode and decode the numbers, which in turn can be used in military base, ATM and so on.

Bilangan Invers Dominasi Total Pada Triangular Snake Graph, Line Triangular Snake Graph, dan Shadow Triangular Snake Graph

Let G = (V(G), E(G)) be a connected graph, where V(G) is the set of vertices and E(G) is the set of edges. The set Dt(G) is called the total domination set in G if every vertex v 2 V(G) is adjacent to at least one vertex in Dt (G). Furthermore, Dt(G) must satisfy the property N(Dt ) = V(G), where N(Dt) is an open neighbourhood set of Dt(G). Suppose that Dt(G) is the total domination set with minimum cardinality. If V(G) - Dt(G) contains a total domination set Dt-1(G), then Dt-1(G) is the inverse set of total domination relative to the total domination set Dt (G). The inverse’s number of the total domination set denotes the minimum cardinality of the inverse set of total domination. This number is denoted by gt-1 (G). This article discusses the inverse’s number of total domination of the triangular snake graph (Tn), line triangular snake graph (L (Tn)), and shadow triangular snake graph (D2 (Tn)). Graph Tn is a graph obtained from the path graph (Pn) by replacing each side of the path with a cycle graph (C3). Graph L (Tn) is a graph where the vertex set in L(Tn) is the edge set on Tn, or V(L(Tn)) = E(Tn). Graph D2 (Tn) is a graph obtained by combining two copies of a graph Tn, namely Tn0 and T00n. This research shows that the graph Tn does not have an inverse of domination total, gt-1 (L (Tn)) = n for n = 4, 6, 8, gt-1 (L (Tn)) = n - 1 for n = 3, 5, 7, or n ≥ 9 with n 2 N, and gt-1 (D2 (Tn)) = b23nc for n ≥ 3 with n 2 N.

Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs

&lt;abstract&gt;&lt;p&gt;In this paper, we study depth and Stanley depth of the quotient rings of the edge ideals associated to triangular and multi triangular snake and triangular and multi triangular ouroboros snake graphs. In some cases, we find exact values, otherwise, we find tight bounds. We also find lower bounds for the edge ideals of triangular and multi triangular snake and ouroboros snake graphs and prove a conjecture of Herzog for all edge ideals we considered.&lt;/p&gt;&lt;/abstract&gt;

Upper Bounds of Radio Number for Triangular Snake and Double Triangular Snake Graphs

A radio labeling of a simple connected graph G = (V, E) is a function h : V⟶N such that |h(x) − h(y)| ≥ diam(G) + 1 − d(x, y), where diam (G) is the diameter of graph and d(x, y) is the distance between the two vertices. The radio number of G, denoted by rn (G), is the minimum span of a radio labeling for G. In this study, the upper bounds for radio number of the triangular snake and the double triangular snake graphs are introduced. The computational results indicate that the presented upper bounds are better than the results of the mathematical model provided by Badr and Moussa in 2020. On the contrary, these proposed upper bounds are better than the results of algorithms presented by Saha and Panigrahi in 2012 and 2018.

Edge odd graceful of alternate snake graphs

Let G be a graph with vertex set V(G), edge set E(G), and the number of edges q. An edge odd graceful labeling of G is a bijection f : E(G) → {1,3,5, …,2q − 1} so that induced mapping f + : V(G) → {0,1,2, …,2q − 1} given by f +(x) = ∑ xy∈E(G) f(xy) (mod 2q) is injective. A graph which admits an edge odd graceful labeling is called edge odd graceful. An alternate triangular snake graph is a graph obtained from a path u 1 u 2 u 3 … u 2m by joining every u 2i−1 and u 2i to a new vertex υi , 1 ≤ i ≤ m. An alternate quadrilateral snake graph is a graph obtained from vertices u 1, u 2,u 3,…,u 2m by joining every u 2i−1 and u 2i to two vertices υi and wi ,1 ≤ i ≤ m, and joining every u 2i to u 2i+1 with 1 ≤ i ≤ m − 1. In this paper, we show that alternate triangular snake and alternate quadrilateral snake graphs are edge odd graceful.

Sigma chromatic numbers of the middle graph of some families of graphs

Let G be a nontrivial connected graph and let c : V (G) → ℕ be a vertex coloring of G, where adjacent vertices may have the same color. For a vertex υ of G, the color sum σ(υ) of υ is the sum of the colors of the vertices adjacent to υ. The coloring c is said to be a sigma coloring of G if σ(u) ≠ σ(υ) whenever u and υ are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). In this study, we investigate sigma coloring in relation to a unary graph operation called middle graph. We will show that the sigma chromatic number of the middle graph of any path, cycle, sunlet graph, tadpole graph, ladder graph, or triangular snake graph is 2 except for some small cases. We also determine the sigma chromatic number of the middle graph of stars.

ON HOP DOMINATION NUMBER OF SOME GENERALIZED GRAPH STRUCTURES

A subset \( H \subseteq V (G) \) of a graph \(G\) is a hop dominating set (HDS) if for every \({v\in (V\setminus H)}\) there is at least one vertex \(u\in H\) such that \(d(u,v)=2\). The minimum cardinality of a hop dominating set of \(G\) is called the hop domination number of \(G\) and is denoted by \(\gamma_{h}(G)\). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.