Tadpole Graph Research Articles

A Study on Zagreb Indices of Vertex-Switching for Special Graph Classes

Many graph theorists have studied graph operations due to their applications and the advantages with heavy calculations. In a recent paper, the vertex-switching operation is analyzed, and some vertex-switched graphs are determined for some graph classes. This paper calculates the first Zagreb index and the second Zagreb index of vertex-switched star, complete bipartite, and tadpole graphs. It finally discusses the need for further research.

On the Laplacian index of tadpole graphs

Abstract In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle C k {C}_{k} and a path P n {P}_{n} . Using two different approaches, we show that their Laplacian index converges to Δ 2 Δ − 1 = 9 2 \frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2} as n , k → ∞ n,k\to \infty , where Δ = 3 \Delta =3 is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic, Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory, J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.

Palindromic Antimagic Labeling of Products of Paw and Banner Graph

This article presents a novel classification of Antimagic labeling referred to as Palindromic antimagic. Palindromic Antimagic labeling pertains to the assignment of palindromic numbers {℘1, ℘2, ℘3 . . . ℘q} to the set of edges of a graph G = (V, E), where G consists of p vertices and q edges. This labeling is characterized by being invertible, meaning that each edge is uniquely associated with a palindromic number. Additionally, it involves an injective mapping of vertex labeling, where the sum of incident edges for any given vertex is distinct from one another. Inthis study, we examine the palindromic antimagic labeling of Cartesian and Tensor product of certain Tadpole graphs, such as the paw and banner graph.

Spectral analysis and stabilization of the dissipative Schrödinger operator on the tadpole graph

In the context of the tadpole graph , we examine the damped Schrödinger semigroup . The first step in our analysis involves a meticulous examination of the spectrum and a suitable decomposition of the resolvent's kernel. By doing so, we establish that the generalized eigenfunctions constitute a Riesz basis of a specific subspace of . As a result, we demonstrate that the energy corresponding to the system decays exponentially.

Further results on random walk labelings

In a previous work, we defined and studied random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that each vertex is labeled upon its first visit. In this work, we calculate the number of random walk labelings of several natural graph families: The wheel, fan, barbell, lollipop, tadpole, friendship, and snake graphs. Additionally, we prove several combinatorial identities that emerged during the calculations.

The Secure Metric Dimension of the Globe Graph and the Flag Graph

Let G = (V, E) be a connected, basic, and finite graph. A subset T=u1,u2,…,uk of V(G) is said to be a resolving set if for any y ∈ V(G), the code of y with regards to T, represented by CTy, which is defined as CTy=du1,y,du2,y,…,duk,y, is different for various y. The dimension of G is the smallest cardinality of a resolving set and is denoted by dim(G). If, for any t ∈ V – S, there exists r ∈ S such that S–r∪t is a resolving set, then the resolving set S is secure. The secure metric dimension of 𝐺 is the cardinal number of the minimum secure resolving set. Determining the secure metric dimension of any given graph is an NP-complete problem. In addition, there are several uses for the metric dimension in a variety of fields, including image processing, pattern recognition, network discovery and verification, geographic routing protocols, and combinatorial optimization. In this paper, we determine the secure metric dimension of special graphs such as a globe graph Gln, flag graph Fln, H- graph of path Pn, a bistar graph Bn,n2, and tadpole graph T3,m. Finally, we derive the explicit formulas for the secure metric dimension of tadpole graph Tn,m, subdivision of tadpole graph ST3,m, and subdivision of tadpole graph STn,m.

Stability theory for two-lobe states on the tadpole graph for the NLS equation

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering δ-type boundary conditions at the junction and bound states with a positive two-lobe profile, the main novelty of this paper is at least twofold. Via a splitting eigenvalue method developed by the author, we identify the Morse index and the nullity index of a specific linearized operator around of an a priori positive two-lobe state profile for every positive power; and we also obtain new results about the existence and the orbital stability of positive two-lobe states at least in the cubic NLS case. To our knowledge, the results contained in this paper are the first in studying positive bound states for the NLS on a tadpole graph by non-variational techniques. In particular, our approach has prospect of being extended to study stability properties of other bound states for the NLS on a tadpole graph or on other non-compact metric graph such as a looping edge graph, as well as, for other nonlinear evolution models on a tadpole graph.

On r-dynamic Coloring of Ladder Graph and Tadpole Graph Using m-shadow Operation

An r-dynamic coloring is a proper k-coloring of a graph G = {V,E} such that the neighbors of every vertex v ∈ V(G) is colored using ς: V(G) → S(c) where S(c) is a set of colors. The coloring is made in such a way that it satisfies the conditions: (i) For any edge uv ∈ E(G), the color of u and color of v are distinct and (ii) The cardinality of coloring the neighbors of any vertex v should be greater than or equal to min{r, d(vG)}, where d(vG) is the degree of the vertex v. In this paper the lower bounds for the r-dynamic coloring of m-shadow graph of ladder graph Dm(Ln) and tadpole graph Dm(Ln,p) are attained. Using the lower bounds the exact solution of r-dynamic chromatic number of the ladder graph Ln and tadpole graph Ln,p by m-shadow operation are obtained.

Bilangan Kromatik Permainan Graf Ubur-Ubur, Graf Siput, dan Graf Gurita

Graph coloring is the process of assigning colors to the vertices or edges of a graph. Specifically, coloring the vertices in graph coloring can be implemented in graph coloring games. This article aims to determine the game chromatic number of the jellyfish graph Jm,n, snail graph SIn, and octopus graph On. For example, give G as a simple, connected, and undirected graph and give two players, namely A as the first player and B as the secondary player. The two players A and B color all the vertices of graph G with available colors. The game’s rules are that A must ensure that all vertices of graph G are colored, while B aims to prevent graph G from being uncolored. Players A and B take turns coloring the vertices of graph G, ensuring that the color of neighboring vertices must be different, with A taking the first turn. If all vertices have been colored, A wins the game, but A loses if some vertices remain uncolored despite available colors, A loses. The smallest value of k for which A has a winning strategy in the game with k colors is the game chromatic number, denoted as χg(G). This thesis discusses the graph coloring game of the tadpole graph Tm,n, broom graph Bn,d, jellyfish graph Jm,n, and tribune graph Tn to find the game chromatic number. The results show that player A uses a strategy to color the vertex with the highest degree in the graph, ensuring that player A always wins. Therefore, the game chromatic number of the jellyfish graph, snail graph, and octopus graph is χg (Jm,n) = 3 for m, n ≥ 1, and χg (SIn) = 3 for n ≥ 1, while χg (On) = 3 for n = 2, 3, 4; χg (On) = 4 for n ≥ 5.

The Effect of Vertex and Edge Removal on Sombor Index

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed from a graph. This is done for several graph elements, including a vertex, an edge, a cut vertex, a pendant edge, a pendant path, and a bridge in a simple graph. Also, pendant and non-pendant cases are studied. Using the obtained formulae successively, one can find the Sombor index of a large graph by means of the Sombor indices of smaller graphs that are just graphs obtained after removal of some vertices or edges. Sometimes, using iteration, one can manage to obtain a property of a really large graph in terms of the same property of many other subgraphs. Here, the calculations are made for a pendant and non-pendant vertex, a pendant and non-pendant edge, a pendant path, a bridge, a bridge path from a simple graph, and, finally, for a loop and a multiple edge from a non-simple graph. Using these results, the Sombor index of cyclic graphs and tadpole graphs are obtained. Finally, some Nordhaus–Gaddum type results are obtained for the Sombor index.

Numerical and Scientific Investigation of Some Molecular Structures Based on the Criterion of Super Classical Average Assignments

Numbering a graph is a very practical and effective technique in science and engineering. Numerous graph assignment techniques, including distance-based labeling, topological indices, and spectral graph theory, can be used to investigate molecule structures. Consider the graph G, with the injection Ω from the node set to 1,2,…,∆, where ∆ is the sum of the number of nodes and links. Assume that the induced link assignment Ω∗ is the ceiling function of the average of root square, harmonic, geometric, and arithmetic means of the vertex labels of the end vertices of each edge. If the union of range of Ω of the node set and the range of Ω of the link set is the set 1,2,…,∆, then Ω is called a super classical average assignment (SCAA). This is known as the SCAA criterion. In this study, the graphical structures corresponding to chemical structures based on the SCAA criterion are demonstrated. The graphical depiction of chemical substances was first defined and second, the union of any number of cycles Cn, the tadpole graph, the graph extracted by identifying a line of any two cycles Cm and Cn, and the graph extracted by joining any two cycles by a path are all examined in this work.

Fast winning strategies for Staller in the Maker–Breaker domination game

The Maker–Breaker domination game is played on a graph G by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex v∈V(G). Dominator’s fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant γSMB′(G) (resp., γSMB(G)) which is the smallest integer k such that, under any strategy of Dominator, Staller can win the game by playing at most k vertices, if Staller (resp., Dominator) plays first on the graph G.We prove some basic properties of γSMB(G) and γSMB′(G) and study the parameters’ changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality δ(G)+1≤γSMB′(G)≤γSMB(G) always holds and that for every three integers r,s,t with 2≤r≤s≤t, there exists a graph G such that δ(G)+1=r, γSMB′(G)=s, and γSMB(G)=t. We prove exact formulas for γSMB′(G) where G is a path or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.

Chemical Applicability of Second Order Sombor Index

In this paper, we introduce the higher-order Sombor index of a molecular graph. In addtition, we compute the second order Sombor index of some standard class of graphs and line graph of subdivision graph of 2D-lattice, nanotube and nanotorus of TUC_4C_8[p,q] and also we obtain the expressions of the second order Sombor index of the line graph of subdivision graph of tadpole graph, wheel graph, ladder graph and chain silicate network CSn. Further, we study the linear regression analysis of the second order Sombor index with the entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization of an octane isomers.

The Metric Dimension of Subdivisions of Lilly Graph, Tadpole Graph and Special Trees

Consider a robot that is navigating in a space represented by a graph and wants to know its current location. It can send a signal to find out how far it is from each set of fixed landmarks. We study the problem of computing the minimum number of landmarks required, and where they should be placed so that the robot can always determine its location. The set of nodes where the landmarks are located is called the metric basis of the graph, and the number of landmarks is called the metric dimension of the graph. On the other hand, the metric dimension of a graph <I>G</I> is the smallest size of a set <I>B</I> of vertices that can distinguish each vertex pair of <I>G</I> by the shortest-path distance to some vertex in <I>B</I>. The finding of the metric dimension of an arbitrary graph is an NP-complete problem. Also, the metric dimension has several applications in different areas, such as geographical routing protocols, network discovery and verification, pattern recognition, image processing, and combinatorial optimization. In this paper, we study the metric dimension of subdivisions of several graphs, including the Lilly graph, the Tadpole graph, and the special trees star tree, bistar tree, and coconut tree.

ON HARMONIOUS CHROMATIC NUMBER OF C(Tm,n), M(Tm,n), C(L(Tm,n) AND M(L(Tm,n))

In this paper, we discuss the harmonious coloring and investigate the harmonious chromatic number of central and middle graph of tadpole graph and harmonious chromatic number of central and middle graph of line graph of tadpole graph, denoted by H(C(Tm,n )), H(M(Tm,n )) and H(C(L(Tm,n ))), H(M(L(Tm,n ))) respectively.

The minimum normalized cut value of tadpole graphs

<p class="Default">The Tadpole graph <em>T_n,k</em>is a Lollipop type graph obtained by joining a one vertex of cycle graph <em>C_n </em>to the end vertex of a path graph <em>P_k</em> The normalized cut is a measure of disassociation between two groups which computes the cut cost as a fraction of the total edge connections to all the vertices in the graph. This research focuses on deriving a formula to find the minimum normalized cut value of Tadpole graphs.

On sombor index of m-shadow graph of some graphs

This new topological invariant is applied in the areas of chemical graph theory. In this paper we have obtained the Sombor index of m−shadow graphs of path graph Dm(Pn), cycle graph Dm(Cn), ladder graph Dm(Ln) and tadpole graph Dm(Tn,p). We have also concluded that, the Sombor index of m-shadow graph of any regular or finite connected graph G is SO(G)m3

Applying the (1,2)-pitchfork domination and it's inverse on some special graphs

Let G be a finite simple and undirected graph without isolated vertices. For any non-negative integers j and k, a subset D of V is called a pitchfork dominating set if every vertex in D dominates at least j and at most k vertices of V - D. A subset D -1 of V - D is an inverse pitchfork dominating set if it is a dominating set. The pitchfork domination number of G, denoted by pf (G) is a minimum cardinality over all pitchfork dominating sets in G. The inverse pitchfork domination number of G, denoted by pf-1 (G) is a minimum cardinality over all inverse pitchfork dominating sets in G. In this paper, pitchfork dominations and it's inverse are applied when j = 1 and k = 2 on some standard graphs such as: tadpole graph, lollipop graph, lollipop flower graph , daisy graph and Barbell graph.

Tadpole domination number of graphs

The graph obtained by joining cycle [Formula: see text] to a path [Formula: see text] with a bridge is called Tadpole graph denoted by [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is said to be a tadpole dominating set of [Formula: see text] if [Formula: see text] is a dominating set and the set of vertices of [Formula: see text] spans a tadpole graph [Formula: see text] where [Formula: see text], [Formula: see text]. In this paper, we find the tadpole domination number of cartesian product of certain graphs, namely, paths, cycles and complete graphs. Also the tadpole domination number for the Mycielskian of cycles and paths is obtained.

On prime labeling of union of tadpole graphs

A graph $G$ of order $n$ is said to be a prime graph if its vertices can be labeled with the first $n$ positive integers in such a way that the labels of any two adjacent vertices in $G$ are relatively prime. If such a labeling on $G$ exists then it is called a prime labeling. In this paper we seek prime labeling for union of tadpole graphs. We derive a necessary condition for the existence of prime labelings of graphs that are union of tadpole graphs and further show that the condition is also sufficient in case of union of two or three tadpole graphs.