Gear Graph Research Articles

A STUDY ON ZERO-M CORDIAL LABELING

A labeling $f: E(G) \rightarrow \{1, -1\}$ of a graph G is called zero-M-cordial, if for each vertex v, the arithmetic sum of the labels occurrence with it is zero and $|e_{f}(-1) - e_{f}(1)| \leq 1$. A graph G is said to be Zero-M-cordial if a Zero-M-cordial label is given. Here the exploration of zero - M cordial labelings for deeds of paths, cycles, wheel and combining two wheel graphs, two Gear graphs, two Helm graphs. Here, also perceived that a zero-M-cordial labeling of a graph need not be a H-cordial labeling.

Vertex irregular reflexive labeling of disjoint union of gear and book graphs

We define [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text]. The labeling [Formula: see text] is called a vertex irregular reflexive [Formula: see text]-labeling if any vertices have distinct weight which [Formula: see text] for any vertices in a graph [Formula: see text]. The weight of a vertex [Formula: see text] is defined as the sum of the labels of vertex and the labels of all edges incident this vertex. The smallest [Formula: see text] for which such labeling exists is called reflexive vertex strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the reflexive vertex strength of gear graphs, book graphs, triangular book graph and the disjoint union of gear graphs.

Double Domination Number of Some Families of Graph

In a graph G = (V, E) each vertex is said to dominate every vertex in its closed neighborhood. In a graph G, if S is a subset of V then S is a double dominating set of G if every vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double domination number γx2 (G). [4]. In this paper, we computed some relations between double domination number, domination number, number of vertices (n) and maximum degree (Δ) of Helm graph, Friendship graph, Ladder graph, Circular Ladder graph, Barbell graph, Gear graph and Firecracker graph.

Root quad mean labeling for some special graphs

A graph G $= (p,q)$ is a Root Quad Mean graph if there is an injective function f from the vertices of G to 1,2,…, $p^2 -1$ such that when each edge uv is labeled with $\frac{\sqrt{f(u)^4+f(v)^4}}{2}$ then the resultant edges are distinct. In this paper we proved Comb graph, Ladder graph , Slanting ladder graph, Gear graph, Helm graph and Heger graph are Root Quad Mean Graphs.

Local super antimagic total vertex coloring of some wheel related graphs

Let G be a simple and finite graph with vertex set V and edge set E. The local antimagic total labeling of G is a map from V ∪ E to the set of positive integers from 1 to |V| + |E| such that the weight of two adjacent vertices are distinct. The weight of a vertex v is calculated by the sum of the label of the vertex v and the labels of all edges incident to it. If the vertices of G is labelled by the smallest labels, that is, {1, 2,…, |V|}, then the such labeling is called local super antimagic total labeling. Thus, the local super antimagic total labeling induces a proper vertex coloring of G where the vertex v is colored by the weight of vertex v. The minimum number of colors taken over all colorings induced by super local antimagic total labeling of G, is called local super antimagic total chromatic number of graph G, and denoted by χlsat (G). In this paper, we consider the local super antimagic total chromatic number of some wheel related graphs such as fans, even gear graphs, and sun flower graphs.

WHEEL RELATED ONE POINT UNION OF VERTEX PRIME GRAPHS AND INVARIANCE

We investigate one point unions of W4, and graphs obtained from W4 such as gear graph G4, each cycle edge of W4 replaced with Pm, each pokes of W4 replaced with Pm for vertex prime labeling. All different non isomorphic structures of these graphs obtained by taking one point union graphs are shown to be vertex prime. This property of graphs is called as invariance under vertex prime labeling.

Edge δ− Graceful Labeling for Some Cyclic-Related Graphs

In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.

Two distance forcing number of a graph

Motivated from the graph parameters namely zero forcing number, k-forcing number and the connected k-forcing number, in this article, we introduce a new parameter known as the 2-distance forcing number. Assume that each vertex of a graph G = (V(G),E(G)) is colored as either white or black. Consider the set Z2d of black colored vertices of the graph G. The color change rule changes the color of a white vertex v to black if the white vertex v is the only 2-distance white neighbor of a black vertex u. The set Z2d is called a two distance forcing set of G if all vertices of the graph G will be turned black after limited applications of the color change rule. The 2-distance forcing number of G, denoted by Z2d(G), is the minimum of |Z2d| over all 2- distance forcing sets Z2d ⊆ V(G). This manuscript is intended to study the 2-distance forcing number of some graphs. We find the exact value of the 2-distance forcing number of graphs such as the pineapple graph, gear graph, jelly fish graph, helm graph, sunflower graph, comet graph and the n-pan graph.

Radio Geometric graceful graphs

A one-one mapping f : V (G) → Z+ satisfying the condition , for every pair of distinct vertices in G is defined as radio geometric mean labeling of G. The maximum number assigned to any vertex of G under the labeling f is called its radio geometric mean number of f denoted rgmn(f). The least value of rgmn(f), taken over all such labelings f of G is defined as its radio geometric mean number and is denoted by rgmn(G). Clearly, rgmn(G) ≥ |V(G)|. Graphs for which rgmn(G) = |V(G)| are defined as radio geometric graceful. In this paper, we find the radio geometric mean number of certain classes of graphs like sunflowers, Helms, gear graphs and show that they are radio geometric graceful.

On Super (a, d) − H− Antimagic Total Labeling of Edge Corona Product and Corona Product of Gear Graph with Path and Cycle

Let graph G consisting of a set of vertices V (G) and a set of edges E(G) be a simple, finite and connected graph. Graph G admits a H− covering, if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. An (a, d) − H− antimagic total labeling of G is a bijective function , such that for all subgraphs H’ that are isomorphic to H. The H− weights φ(H’) = ∑v∈V(H’) ξ(v) + ∑e∈E(H’) ξ(e) constitute an arithmetic progression a, a+d, a+2d, …, a+(k −1)d where a and d are positive integers and k is the number of subgraphs of G isomorphic to H. Moreover, if the smallest possible labels appear on the vertices, then we called G is super (a, d) − H− antimagic. This research aims to study super (a, d) − C4 ʘ Pn antimagic total labeling on Gm ◊ Pn and super (a, d) − C4 ◊ Cn antimagic total labeling on Gm ◊ Cn.

Some New Kulli-Basava Topological Indices

Recently, Kulli-Basava indices were introduced and studied their mathematical and chemical properties which have good response with mean isomer degeneracy. In this paper, we introduce the modified first and second Kulli-Basava indices, F1-Kulli-Basava index, square Kulli-Basava index of a graph, and compute exact formulas for regular graphs, wheels, gear graphs and helm graphs.

On irregular coloring of wheel related graphs

An irregular coloring is a proper coloring in which distinct vertices have different color codes. In this paper, we obtain the irregular chromatic number for middle graph, total graph, central graph and line graph of wheel graph, helm graph, gear graph and closed helm graph.

On the chromatic number local irregularity of related wheel graph

A function f is called a local irregularity vertex coloring if (i) as vertex irregular k-labeling and , for every where and (ii) max(l) = min{max{li};livertex irregular labeling}. The chromatic number of local irregularity vertex coloring of G, denoted by , is the minimum cardinality of the largest label over all such local irregularity vertex coloring. In this article, we study the local irregularity vertex coloring of related wheel graphs and we have found the exact value of their chromatic number local irregularity, namely web graph, helm graph, close helm graph, gear graph, fan graph, sun let graph, and double wheel graph.

On local super antimagic face coloring of plane graphs

Let G(V, E) be a connected graph of order n and size m. A bijection g: V(G) ∪ E(G) → {1,2, …, n + m} is called a local super antimagic face coloring such that for any two adjacent face A1 and A2, w(A1) ≠ w(A2) where w(A) = ∑v∈ V(A)f(v) + ∑e∈ E(A)f(e). The local super antimagic face coloring chromatic number γlaf(G) defined the minimum number of colors taken over all colorings of G induced by local super antimagic face coloring of G. In this paper, we study local super antimagic face coloring of some plane graphs. The name of graphs are gear graph (Jn), prism graph (Pn), double fan graph (Dfn), and antiprism graph (Apn).

Stanley Depth of Edge Ideals of Some Wheel-Related Graphs

Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish general closed formulas for Stanley depth of quotient of edge ideals associated with the m t h -power of a wheel graph, for m ≥ 3 , gear graphs and anti-web gear graphs.

The Complexity of Some Classes of Pyramid Graphs Created from a Gear Graph

The methods of measuring the complexity (spanning trees) in a finite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we defined some classes of pyramid graphs created by a gear graph then we developed the Kirchhoff's matrix tree theorem method to produce explicit formulas for the complexity of these graphs, using linear algebra, matrix analysis techniques, and employing knowledge of Chebyshev polynomials. Finally, we gave some numerical results for the number of spanning trees of the studied graphs.

Star edge coloring of corona product of path and wheel graph families

A star edge coloring of a graph G is a proper edge coloring without bichromatic paths and cycles of length four. In this paper, we obtain the star edge chromatic number of the corona product of path with cycle, path with wheel, path with helm and path with gear graphs, denoted by Pm ◦ Cn, Pm ◦ Wn, Pm ◦ Hn, Pm ◦ Gn respectively.

Mean Square Cordial Labeling of Some Cycle Related Graphs

In this paper we investigate the Mean square cordial labeling for some cyclic graphs like Helm graph Hp, Closed helm graph CHp, Gear graph Gp, Sunlet graph SLp, Fan graph F1,p and CpʘnK1 .

Edge Irregular Reflexive Labeling for the Disjoint Union of Gear Graphs and Prism Graphs

In graph theory, a graph is given names—generally a whole number—to edges, vertices, or both in a chart. Formally, given a graph G = ( V , E ) , a vertex naming is a capacity from V to an arrangement of marks. A diagram with such a capacity characterized defined is known as a vertex-marked graph. Similarly, an edge naming is a mapping of an element of E to an arrangement of marks. In this case, the diagram is called an edge-marked graph. We consider an edge irregular reflexive k-labeling for the disjoint association of wheel-related diagrams and deduce the correct estimation of the reflexive edge strength for the disjoint association of m copies of some wheel-related graphs, specifically gear graphs and prism graphs.

WALK POLYNOMIAL: A New Graph Invariant

Abstract In this paper we introduce the walk polynomial to find the number of walks of different lengths in a simple connected graph. We also give the walk polynomial of the bipartite, star, wheel, and gear graphs in closed forms.