Pendant Vertices Research Articles

The spectral Turán problem about graphs of given size with forbidden subgraphs

Let G be a graph of size m and ρ ( G ) be the spectral radius of its adjacency matrix. A graph is said to be H-free if it does not contain a subgraph isomorphic to H. Let C l 1 be the graph obtained by adding a pendent vertex to a cycle of length l. In this paper, we determine the graphs with the maximum spectral radius over C 3 1 -free graphs, and C 4 1 -free graphs with m ≥ 9 .

Analyzing the modified symmetric division deg index: mathematical bounds and chemical relevance

Abstract The modified symmetric division deg (MSD) index of a graph G is precisely defined as MSD ( G ) = ∑ j ∼ k 1 2 d j d k + d k d j , where d j and d k represent the degrees of j and k respectively. In this paper, we present precise bounds for the modified symmetric division deg index, expressed in the relation to the minimum and maximum degrees, the order and size of the graph, the number of pendant vertices and the minimum degree of a non-pendant vertex, the forgotten topological index, the modified second Zagreb index. We determine the upper bounds for the modified symmetric division deg index of unicyclic, bicyclic and k-cyclic graphs. Moreover, upon analyzing the modified symmetric division deg index, we observe its correlation with other well-known indices and its chemical applicability on the molecular graphs of octane isomers. At the end, we find the chemical applicability of the modified symmetric division deg index on benzenoid hydrocarbons and observe that the modified symmetric division deg index has a very strong correlation with the physical properties of benzenoid hydrocarbons.

The eigenvalue multiplicity of line graphs

Let m(G,λ), c(G) and p(G) be the multiplicity of an eigenvalue λ, the cyclomatic number and the number of pendant vertices of a connected graph G, respectively. Yang et al. (2023) [10] proved that m(L(T),λ)≤p(T)−1 for any tree T, and characterized all trees T with m(L(T),λ)=p(T)−1, where L(T) is the line graph of T. In this paper, we extend their result from a tree T to any graph G≠Cn, and prove that m(L(G),λ)≤2c(G)+p(G)−1 for any graph G≠Cn. Moreover, all graphs G with m(L(G),−1)=2c(G)+p(G)−1 are completely characterized.

Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with k Pendant Vertices

Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with k Pendant Vertices

On the maximum $$A_{\alpha }$$-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

On the maximum $$A_{\alpha }$$-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

Extremal graph of super line graph operation via generalized Randić index

Graph operations enable the modification and transformation of graphs, enhancing data representation as well as effectiveness in fields such as computer science, mathematics, and data analysis. Super line graph operations are a concept of advanced graph theory that is used in advanced mathematics and computer science. They are used to solve specific mathematical problems, especially in graph theory and Combinatorics. Topological numbers are numerical values connected with graph structures that are used to analyze structural characteristics and solve problems in fields such as chemistry and mathematics. They are important for simplifying complex data and enabling quantitative analysis. The general Randić number of some super line graph operations of number two with a three-diameter is investigated in this study. The pendant vertices are inserted without disturbing their diameter. Seven graphs of order five, each with a diameter of three, can be used to create twenty-three generalized super line graphs.

Exploring topological indices of oligothiophene dendrimer via neighbourhood M-polynomial

In this study, we give some theoretical results for studying the structural connectivity of oligothiophene dendrimer by utilizing the neighborhood M-polynomial. The main result of this article presents the closed formula of neighborhood M-polynomial of oligothiophene dendrimer which is used for computing the neighborhood topological indices of the dendrimer. Among the various indices considered, the forgotten index obtained the highest value, followed by the Sanskruti index, first Zagreb index, and second Zagreb index. These topological indices exhibited an increasing trend as the number of generation increases across the dendrimer structure. Conversely, the remaining index values display minimal or negligible changes as the generation increases. This suggests that these indices exhibits slow increases with higher generations in the dendrimer structure. The significance of adopting the neighborhood M-polynomial to compute the topological index lies in the fact that it facilitates the analysis of complex molecules in a timely manner and allows pendant vertex to contribute more effectively to the overall degree value of the dendrimer. Furthermore, the dataset resulting from these topological indices serves as a foundation for future studies aimed at predicting the physicochemical properties of the dendrimer.

Exploration of CCD-Number for Some Specific Graphs

Objectives: A dominating set S of a graph G is said to be a complementary corona dominating set (CCD-set) if every vertex in is either a pendent vertex or a support vertex. The smallest cardinality of a CCD-set is called the CCD-number and is denoted by . In this article, we explore the CCD-number for some specific graphs. Methods: We are finding the upper bound of and the lower bound of . We prove that both the upper and lower bound are equal for the given graphs. Findings: In this article, we obtained the CCD-number for some specific graphs. Novelty: Complementary corona dominating set is a vertex set such that the induced subgraph of the complementary set is isomorphic to corona virus. Keywords: Dominating Set, Corona Dominating Set, Support And Pendent Vertex, Butterfly Graph, Dumbbell Graph, Coconut Tree Graph, Peacock Head Graph, Sunlet Graph, Spider Graph, Firecracker Graph, Uniform g-ply graph

Outer Triple Connected Corona Domination Number of Graphs

Background/ Objective: Given a graph G, a dominating set is said to be corona dominating set if every vertex such that or there exist a vertex if then . A corona dominating set is said to be an outer triple connected corona dominating set if any three vertices in lie on a path. The minimum cardinality taken over all the outer triple connected corona dominating sets of is called outer triple connected corona dominating number and it is denoted by . The study aims to find the outer triple connected corona domination number of some graphs. Method: To obtain outer triple connected corona domination number say m by proving and . To prove for a graph G we find a outer triple connected corona dominating set of G with cardinality m and then to prove we prove by contradiction. Findings: We investigated the above parameter for some derived graphs of path, cycle and wheel graph. Novelty : Outer triple connected corona domination number is a new concept in which the conditions of corona domination and triple connected are linked together. Keywords: Corona Domination, Pendent Vertex, Support Vertex, Triple Connected, Isolated Vertex, Pendant vertes

Generalization of CD-Number for Power Graph of Some Special Types of Tree Graphs

Objectives: The main objective of the article is to finding the corona domination for the power graph of some special types of tree graph. Method: A dominating set of a graph is said to be a corona dominating set if every vertex in is either a pendant vertex or a support vertex. The minimum cardinality of a corona dominating set is called the corona domination number and is denoted by . Findings: In this article, we study the -number for the power of PVB-tree and where and identify their exact values. Novelty: The corona domination was one of the recently developed domination parameter, along with this parameter we found the exact value for some special types of graphs. Keywords: Domination Number, Corona Domination Number, Total Domination, Power Graphs, Olive Tree And PVB­Tree

Eigenvalue multiplicity of a graph in terms of the number of external vertices

The multiplicity of an eigenvalue λ of a graph G is denoted by m(G,λ). In a connected graph G with at least two vertices, a vertex is called external if it is not a cut vertex. In a tree, an external vertex is exactly a pendant vertex. Let ϵ(G) be the number of external vertices in G. In this paper, we prove that m(G,λ)≤ϵ(G)−1 for any λ and characterize the extremal graphs with m(G,−1)=ϵ(G)−1, which generalizes the main result of Wang et al. [Linear Multi. Alg., 2020] from a tree to an arbitrary connected graph.

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

The T1,2-free planar graphs whose second largest eigenvalue does not exceed 1

The problem of characterizing graphs whose second largest eigenvalue does not exceed a comparatively small constant has attracted a great deal of attention in the last decades. Let T1,2 be the tree of diameter 3 with exactly 3 pendent vertices. In this paper, all connected T1,2-free planar graphs whose second largest eigenvalue does not exceed 1 are determined.

On the maximum atom-bond sum-connectivity index of graphs

Abstract The atom-bond sum-connectivity (ABS) index of a graph G G with edges e 1 , … , e m {e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1 − 2 ( d e i + 2 ) − 1 \sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1 ≤ i ≤ m 1\le i\le m , where d e i {d}_{{e}_{i}} is the number of edges adjacent to e i {e}_{i} . In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

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.

Restrained captive domination number

Abstract The restrained captive domination number (RCDN), denoted by γ Rca ( G ) , {\gamma }_{{\rm{Rca}}}(G), is a new definition of domination number in graphs introduced in this article. If D D is a captive dominating set and G [ V − D ] G{[}V-D] has no isolated vertex, then D D is a restrained captive dominating set (RCDS) of graph G G . Furthermore, some properties, theorems, and propositions are determined. The inverse RCDN is introduced too. The RCDN in complement graphs is discussed. Finally, the RCDS of some graphs is determined. We have explained that if the graph has pendent vertex, then it has no RCDN.

Minimizing the least eigenvalue of unbalanced signed unicyclic graphs with given girth or pendant vertices

A signed graph $\Gamma=(G,\sigma)$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma:E\rightarrow\{1,-1\}$. Let $A(\Gamma)$ be the adjacency matrix of $\Gamma$. Let $\lambda_1(A(\Gamma))\geq\lambda_2(A(\Gamma))\geq\cdots\geq\lambda_n(A(\Gamma))$ be the spectrum of the signed graph $\Gamma$, where $\lambda_n(A(\Gamma))$ is the least eigenvalue of $\Gamma$. Let $\mathcal{U}^-_{n,g,k}$ denote the set of all the unbalanced signed unicyclic graphs with order $n$, girth $g$ and $k$ pendant vertices, let $\mathcal{U}^-_n(k)$ denote the set of all the unbalanced signed unicyclic graphs with $n$ vertices and $k$ pendant vertices, and let $\mathcal{U}^-_{n,g}$ denote the set of all the unbalanced signed unicyclic graphs with order $n$ and girth $g$. Obviously, $\mathcal{U}^-_n(k)=\bigcup\limits_{g=3}^{n-k}\mathcal{U}^-_{n,g,k}$ and $\mathcal{U}^-_{n,g}=\bigcup\limits_{k=0}^{n-g}\mathcal{U}^-_{n,g,k}$. In this paper, we determine the signed unicyclic graphs whose least eigenvalues are minimal among all the graphs in $\mathcal{U}^-_{n,g,k}$, $\mathcal{U}^-_n(k)$ and $\mathcal{U}^-_{n,g}$, respectively.

Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices

For a connected graph G, we denote by mG(μ), c(G) and p(G) the eigenvalue multiplicity of μ in G, the cyclomatic number and the number of pendant vertices in G, respectively. In 2020, Wang et al. (2020) proved that mG(μ)≤2c(G)+p(G) for any μ∈R, the equality holds if and only if G is a cycle and μ is a multiple eigenvalue of the cycle. We find that when ∥μ∥≥2, the bound 2c(G)+p(G) is much rough. Thus we are motivated to improve the bound of mG(μ) from 2c(G)+p(G) to c(G)+q(G) for the case when ∥μ∥≥2, where q(G) is the number of quasi-pendant vertices of G.

Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs.

Sharp Lower Bound of Cacti Graph with respect to Zagreb Eccentricity Indices

The first Zagreb eccentricity index E1(℧) is the sum of square of eccentricities of the vertices, and the second Zagreb eccentricity index E2(℧) is the sum of product squares of the eccentricities of the vertices. A linked graph G is called a cactus if any two of its cycles share only one vertex. In other words, there are no two independent cycles that share an edge. Cactus graphs are also known as “block graphs” or “sensitized graphs.” They are closely related to chordal graphs and can be used to represent various types of networks, including communication networks and road networks. In this contribution, E1(℧) and E2(℧) values of cacti with k pendant vertices and k cycles, respectively, are considered. We determine the minimum E1, E2 indices for n order cacti with k pendant vertices and k cycles.