Helm Graph Research Articles

On locating-chromatic number of helm graph H_m FOR 10≤m≤28

Let G = (V,E) be a connected graph and c be a k-coloring of G. The color class S_i of G is a set of vertices given color i, for 1 ≤ i ≤ k. Let  = {S_1,S_2,…,S_k} be an ordered partition of V(G). The color code of a vertex v $in$ (element) V(G) is defined as the ordered k--tuplec_Π (v)=(d(v,S_1),d(v,S_2),...,d(v,S_k)),where d(v,S_i) = min{d(v,x)| x $in$ (element) S_i} for 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number χ_L (G) is the minimum number of colors in a locating-coloring of G. This paper discusses the locating-chromatic number of helm graph H_m for 10 ≤ m ≤ 28. Helm graph H_m is constructed by adding some leaves to the corresponding vertices of wheels W_m, for m ≥ 3.

On Radio k – Chromatic Number for Mycielski of Some Graphs

Objectives: In radio networks, the main difficulty is managing the radio spectrum, assigning radio frequencies to transmitters optimally without any interferences. This study aims to find the smallest span of Mycielski of some graphs, the maximum color assigned to any node is called span. Methods : This study focused on the problem of reducing interference by modeling it with a radio k - coloring problem on graphs. Where transmitters are modeled as nodes in a graph, with edges connecting nodes that represent transmitters in close proximity to one another. For a graph , with node set , edge set and an integer , a radio k - coloring of is a function satisfying the condition for any two nodes and , where is the distance between and in . The radio k - coloring of , is the maximum color assigned to any node of and it is denoted by . The radio k -chromatic number of is the minimum value of taken over all radio k - coloring of and it is denoted by . Findings: This study obtained the radio k-chromatic number for Mycielski of some graphs for and 3 Novelty: To solve the channel assignment problem in radio transmitters, the interference graph is developed, and the channel assignment has been converted into a graph coloring. Reducing the interference by a radio k - coloring problem will motivate many researchers to find the radio k - coloring in various graphs. Keywords: Radio k – Coloring, Mycielski graph, Double Star graph, Triple Star graph, Sunlet graph, Helm graph and Closed Helm graph

Fuzzy Range Labeling of Fuzzy Star Graph and Helm Graph

Objectives: To find a new labeling technique in Fuzzy Graph Theory called Fuzzy Range Labeling. This concept helps to handle the uncertainty in graph labeling areas. Methods: Fuzzy graph labeling condition - and Range labeling condition - (that is, the edge membership value is calculated as the difference between the maximum and minimum values of the vertex incident to that edge) are used. Findings: In this study, we have found that the fuzzy star graph and helm graph are fuzzy range graphs. Novelty: The fuzzy range labeling technique was not yet studied. Thus, this new notion will motivate many researchers to study further. Keywords: Fuzzy graph labeling, Range labeling, Fuzzy range labeling, Fuzzy range graph, Fuzzy star graph, Helm graph

Induced Topologies of Independent Domination in Helm Graphs

Let G be a graph. [1] The independent domination topology (ID topology) of G, denoted by TI (G) is the topology generated by the family IG of all independent dominating sets of G. In this paper, we introduce and characterize and describe the independent domination topology through the context of the independent dominating sets of the helm graph Hn. This study highlights the significance of this topology in optimizing network designs and computational systems, offering a foundation for future research and practical applications.

On Null Vertex in Bipolar Fuzzy Graphs

We present a novel vertex in Bipolar fuzzy graph, null vertex, which is distinct from boundary vertex and interior vertex and also attempt a study on null vertex in bipolar fuzzy closed helm graph CHn.

Locating Chromatic Number of Middle Graph of Path, Cycle, Star, Wheel, Gear and Helm Graphs

Let \(c\) be a proper \(k\)-coloring of a connected graph \(G\) and \(\pi=\{S_{1},S_{2},\ldots,S_{k}\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(S_{i}\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(c_{\pi}(v)\) of \(v\) with respect to \(\pi\) is the ordered \(k\)-tuple \(c_{\pi}(v)=(d(v,S_{1}),d(v,S_{2}),\ldots,d(v,S_{k}))\), where \(d(v,S_{i})=min\{d(v,u):\textit{ } u\in S_{i}\}\) for \(1\leqslant i \leqslant k\). If all distinct vertices of \(G\) have different color codes, then \(c\) is called a locating coloring of \(G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.

Geodetic Domination Integrity of Thorny Graphs

The concept of geodetic domination integrity is a crucial parameter when examining the potential damage to a network. It has been observed that the removal of a geodetic set from the network can increase its vulnerability. This study explores the geodetic domination integrity parameter and presents general results on the geodetic domination integrity values of thorn ring graphs, $n$-sunlet graphs, thorn path graphs, thorn rod graphs, thorn star graphs, helm graphs, $E_p^t$ tree graphs, dendrimer graphs, spider graphs, and bispider graphs, which are the frequently used graph classes in the literature.

Burning, edge burning & chromatic burning classification of some graph family

Graph ‘G’ is a Simple and undirected graph, which has a lowest number of color that is required to color the edge is called chromatic index. It is denoted by the symbol χ1(G). In order to burn a graph, a minimum source of edges is required which is called edge burning index or edge burning number and it is denoted by the symbol b1(G). A minimum cardinality of color graph set and the source of the burning index set are called edge chromatic burning and let us assume that it is represented by the symbol χb1(G). In this paper we are mainly concentrating on identifying the chromatic burning and edge chromatic burning graphs and their chromatic burning index using a specific graph family. They are Petersen graph, Helm graph and n- Sunlet graph.

The $r$-dynamic edge coloring of a closed helm graph

The $r$-dynamic edge coloring of a closed helm graph

LOCAL IRREGULARITY POINT COLORING ON THE RESULT OF SUBDIVISION OPERATION OF HELM GRAPHS

One of the sub-chapters studied in graphs is local irregularity vertex coloring of graph. The based on definition of local irregularity vertex coloring of graph, as follow : (i)l : V (G) →{1, 2, 3, . . . , k} as a vertex irregular labeling and w : V (G) → N, for every uv ∈ E(G), w(u) ̸=w(v) with w(u) = Pv∈N(u)l(v) and (i) Opt(l) = min{max(li); li is a vertex irregular labeling}. The chromatic number of the local irregularity vertex coloring of G denoted by χlis(G), is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this article, discuss about local irregularity vertex coloring of subdivision by helm graph (Sg(Hn)).

On the local metric dimension of generalized wheel graph

Metric dimension and local metric dimension are distance-based tools that find their applications in robot navigation, combinatorial optimization, pattern recognition, image processing, integer programming, and many more. A study based on the selection of the minimum number of vertices in a graph that will separate all the neighboring vertices in terms of distance between the vertices is given by the local metric basis problem. It is well known that the problem of finding the local metric basis of a graph is NP-complete. In this paper, the local metric dimension of double wheel graph [Formula: see text], generalized wheel graph [Formula: see text], and closed helm graph [Formula: see text] had been computed. It has been found that these graphs have unbounded local metric dimensions.

A Note on Distance Irregular Labeling of Graphs

Let us consider a~simple connected undirected graph G = ( V , E ) . For a~graph G we define a~ k -labeling ϕ : V ( G ) → { 1 , 2 , … , k } to be a~distance irregular vertex k -labeling of the graph G if for every two different vertices u and v of G , one has w t ( u ) ≠ w t ( v ) , where the weight of a~vertex u in the labeling ϕ is w t ( u ) = ∑ v ∈ N ( u ) ϕ ( v ) , where N ( u ) is the set of neighbors of u . The minimum k for which the graph G has a~distance irregular vertex k -labeling is known as distance irregularity strength of G , it is denoted as d i s ( G ) . In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order 1 , friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

A Note on Distance Irregular Labeling of Graphs

Let us consider a~simple connected undirected graph \(G=(V,E)\). For a~graph \(G\) we define a~\(k\)-labeling \(\phi: V(G)\to \{1,2, \dots, k\}\) to be a~distance irregular vertex \(k\)-labeling of the graph \(G\) if for every two different vertices \(u\) and \(v\) of \(G\), one has \(wt(u) \ne wt(v),\) where the weight of a~vertex \(u\) in the labeling \(\phi\) is \(wt(u)=\sum\limits_{v\in N(u)}\phi(v),\) where \(N(u)\) is the set of neighbors of \(u\). The minimum \(k\) for which the graph \(G\) has a~distance irregular vertex \(k\)-labeling is known as distance irregularity strength of \(G,\) it is denoted as \(dis(G)\). In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order \(1,\) friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

Embedding Complete Bipartite Graphs into Wheel Related Graphs

This study considers the exact wirelength of embedding complete bipartite graphs into wheel related graphs such as wheel graphs, gear graphs, and helm graphs.

The Moore-Penrose inverse of the distance matrix of a helm graph

In this paper, we give necessary and sufficient conditions for a real symmetric matrix and, in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank-one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semi-definite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\mathbf{w}\in \mathbb{R}^{2n-1}$ such that\begin{eqnarray*}D(H_n)^{\dagger} = -\frac{1}{2}L +\frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}},\end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.

Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA

Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints.

A note on m-Zumkeller cordial labeling of graphs

Let G = (V,E) be a graph. An m-Zumkeller cordial labeling of the graph G is defined by an injective function f : V → N such that there exists an induced function f ∗ : E → {0, 1} defined by f ∗ (uv) = f (u) · f (v) that satisfies the following conditions- i. For every uv ∈ E, f ∗ (uv) = {█(1,if f (u) · f (v) is an m-Zumkeller number @ 0,otherwise )┤ ii. |ef ∗ (0) − ef∗ (1)| ≤ 1 where ef ∗ (0) and ef ∗ (1) denote the number of edges of the graph G having label 0 and 1 respectively under f ∗. In this paper we prove that there exist an m-Zumkeller cordial labeling of graphs in paths, cycles, comb graphs, ladder graphs, twig graphs, helm graphs, wheel graphs, crown graphs and star graphs.

TOTAL CHROMATIC NUMBER OF FUZZY BISTAR GRAPH & FUZZY HELM GRAPH

Fuzzy total chromatic number is the least value of k such that k-fuzzy total coloring exist. In this paper, we discussed the concept of total coloring of graphs to Fuzzy bistar graph and Fuzzy helm graph. Here we define fuzzy chromatic number of the fuzzy Bistar graph and Fuzzy Helm graph.

On local edge antimagic chromatic number of graphs

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection f : V → {1, 2, 3, ..., p} is called a local edge antimagic labeling if for any two adjacent edges e = uv and e’ = vw of G, we have w(e) ≠ w(e’), where the edge weight w(e = uv) = f(u)+f(v) and w(e’) = f(v)+f(w). A graph G is local edge antimagic if G has a local edge antimagic labeling. The local edge antimagic chromatic number χ’lea(G) is defined to be the minimum number of colors taken over all colorings of G induced by local edge antimagic labelings of G. In this paper, we determine the local edge antimagic chromatic number for a friendship graph, wheel graph, fan graph, helm graph, flower graph, and closed helm.

Reconstruction of helm graph and web graph

The graph reconstruction conjecture asserts that every simple undirected graph with number of vertices (where <em>n</em> ≥ 3) is uniquely determined up to isomorphism, by its collection of unlabeled form of vertex deleted sub graphs. If a vertex deleted sub graph of <em>G</em> is given in unlabeled form, then it is called a “card” of <em>G</em>. The collection of all cards of <em>G</em> is called the “deck” of <em>G</em>, and is denoted by <em>D</em>(<em>G</em>) . If we can determine the original graph <em>G</em> from the deck of the graph <em>G</em>, we can say that it is reconstructible. In this paper we propose a new method to reconstruct the helm graph (<em>H_n</em>) and the web graph (<em>W_n</em>), by using the degree sequence of its collection of vertex deleted sub graphs. Helm graph is obtained by adjoining a pendant edge to each node of the cycle of the -wheel graph. Web graph is obtained by joining pendant vertices to the vertices in the outer cycle of the helm graph to form a cycle and again adding pendant vertices to the new cycle. For both graphs we found decks and obtained the degree sequence of each cards. Finally, we reconstructed the original graph using the degree sequences.