Extremal Trees Research Articles

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

Extremal Trees with Respect to Bi-Wiener Index

Extremal Trees with Respect to Bi-Wiener Index

Sharp Upper Bound for Harmonic Index of Trees with Given Total Domination Number

Let \( G=(V,E) \) be a simple connected graph with vertex set \( G \) and edge set \( E \). The harmonic index of graph \( G \) is the value \( H(G)=\sum_{uv\in E(G)} \frac{2}{d_u+d_v} \), where \( d_x \) refers to the degree of \( x \). We obtain an upper bound for the harmonic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound.

The maximum number of maximum generalized 4‐independent sets in trees

AbstractA generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence number is the cardinality of a maximum ‐independent set in . Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order is if is odd, and if is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order is if , and if , and if and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized ‐independent sets in a tree for a general integer . As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order is and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets.

Fixed-Order Chemical Trees with Given Segments and Their Maximum Multiplicative Sum Zagreb Index

Topological indices are often used to predict the physicochemical properties of molecules. The multiplicative sum Zagreb index is one of the multiplicative versions of the Zagreb indices, which belong to the class of most-examined topological indices. For a graph G with edge set E={e1,e2,⋯,em}, its multiplicative sum Zagreb index is defined as the product of the numbers D(e1),D(e2),⋯,D(em), where D(ei) is the sum of the degrees of the end vertices of ei. A chemical tree is a tree of maximum degree at most 4. In this research work, graphs possessing the maximum multiplicative sum Zagreb index are determined from the class of chemical trees with a given order and fixed number of segments. The values of the multiplicative sum Zagreb index of the obtained extremal trees are also obtained.

Modified Zagreb index and total domination in trees

Let [Formula: see text] be a simple connected graph. The modified first Zagreb index is denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and total domination number for trees and characterize the extremal trees attaining the bound.

Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices

<abstract><p>Let $ d_u $ be the degree of a vertex $ u $ of a graph $ G $. The atom-bond sum-connectivity (ABS) index of a graph $ G $ is the sum of the numbers $ (1-2(d_v+d_w)^{-1})^{1/2} $ over all edges $ vw $ of $ G $. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with $ n $ vertices and $ p $ pendent vertices is also addressed; such extremal trees have the maximum degree $ 3 $ when $ n\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.</p></abstract>

Extremal trees with fixed degree sequence for $\sigma$-irregularity

Extremal trees with fixed degree sequence for $\sigma$-irregularity

The Geometric–Arithmetic index of trees with a given total domination number

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. In Vukičević and Furtula (2009), Vukičević defined a new topological index, named geometric–arithmetic index of a graph G and denoted by GA(G), as GAG=∑uv∈E2dudvdu+dv,where du and dv denote the degrees of vertices u and v, respectively. We obtain an upper bound for the geometric–arithmetic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound. Additionally, by a known relation between the geometric–arithmetic and arithmetic–geometric indices, we get a new lower bound for the arithmetic–geometric index.

More on independent transversal domination

A set [Formula: see text] of vertices in a graph [Formula: see text] is called a dominating set if every vertex in [Formula: see text] is adjacent to a vertex in [Formula: see text]. Hamid defined a dominating set which intersects every maximum independent set in [Formula: see text] to be an independent transversal dominating set. The minimum cardinality of an independent transversal dominating set is called the independent transversal domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper we prove that for trees [Formula: see text], [Formula: see text] is bounded above by [Formula: see text] and characterize the extremal trees. Further, we characterize the class of all trees whose independent transversal domination number does not alter owing to the deletion of an edge.

Retracted: On the Extremal Trees for Some Bond Incident Degree Indices with a Fixed Number of Segments

Retracted: On the Extremal Trees for Some Bond Incident Degree Indices with a Fixed Number of Segments

Kemeny's constant and Wiener index on trees

On trees of fixed order, we show a direct relation between Kemeny's constant and Wiener index, and provide a new formula of Kemeny's constant from the relation with a combinatorial interpretation. Moreover, the relation simplifies proofs of several known results for extremal trees in terms of Kemeny's constant for random walks on trees. Finally, we provide various families of co-Kemeny's mates, which are two non-isomorphic connected graphs with the same Kemeny's constant, and we also give a necessary condition for a tree to attain maximum Kemeny's constant for trees with fixed diameter.

Extremal graphs with respect to the total-eccentricity index

In a connected graph [Formula: see text], the distance between two vertices of [Formula: see text] is the length of a shortest path between these vertices. The eccentricity of a vertex [Formula: see text] in [Formula: see text] is the largest distance between [Formula: see text] and any other vertex of [Formula: see text]. The total-eccentricity index [Formula: see text] is the sum of eccentricities of all vertices of [Formula: see text]. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.

The k-Sombor Index of Trees

For a positive real number [Formula: see text], the [Formula: see text]-Sombor index of a graph [Formula: see text], introduced by Réti et al, is defined as [Formula: see text] where [Formula: see text] denotes the degree of the vertex [Formula: see text] in [Formula: see text]. By the definition, [Formula: see text] is exactly the Sombor index of [Formula: see text], while [Formula: see text] is the first Zagreb index of [Formula: see text]. In this paper, for [Formula: see text] we present the extremal values of the [Formula: see text]-Sombor index of trees with some given parameters, such as matching number, the number of pendant vertices, diameter. This generalizes the relevant results on Sombor index due to Chen, Li and Wang ((2002). Extremal values on the Sombor index of trees, MATCH Communications in Mathematical and in Computer Chemistry, 87, 23–49). Handling [Formula: see text] appears to be different for [Formula: see text] in contrast to the case when [Formula: see text]. To demonstrate this, we also characterize the extremal trees with respect to the [Formula: see text] with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.

On the computation of extremal trees of Harmonic index with given edge-vertex domination number

Let [Formula: see text] be vertices of a graph [Formula: see text] with degree of the vertices being [Formula: see text] and [Formula: see text] respectively. First, let us define the weight of the edge [Formula: see text] as twice the value of [Formula: see text] in [Formula: see text]. Let us define [Formula: see text], the harmonic index of the graph [Formula: see text], as the sum obtained by adding the weight assigned to every edge of [Formula: see text]. In this paper, for the class of trees, we shall obtain an upper bound for the harmonic index [Formula: see text] in terms of the edge-vertex domination number and the order of [Formula: see text]. Also, we shall ascertain that the equality is true by characterizing the collection of all extremal trees attaining this bound.

Extremal trees and unicyclic graphs with respect to spectral radius of weighted adjacency matrices with property $$P^{*}$$

Extremal trees and unicyclic graphs with respect to spectral radius of weighted adjacency matrices with property $$P^{*}$$

On diameter $5$ trees with the maximum number of matchings

A matching in a graph is any set of edges of this graph without common vertices. The number of matchings, also known as the Hosoya index of the graph, is an important parameter, which finds wide applications in mathematical chemistry. Previously, the problem of maximizing the Hosoya index in trees of radius $2$ (that is, diameter $4$) of fixed size was completely solved. This work considers the problem of maximizing the Hosoya index in trees of diameter $5$ on a fixed number $n$ of vertices and solves it completely. It turns out that for any $n$ the extremal tree is unique. Bibliography: 6 titles.

Lower bounds on the irregularity of trees and unicyclic graphs

The imbalance of an edge in a graph is defined as the absolute value of the difference of the degrees of its end-vertices. The irregularity of a simple graph G is defined as the sum of the imbalances of all edges of G . It is used as a measure to quantify the deviation of a graph from being regular. In this paper, a sharp lower bound on the irregularity of trees and unicyclic graphs in terms of the maximum degree is established and the corresponding extremal trees and unicyclic graphs are characterized.

Strong restrained domination number on trees and product of graphs: An algorithmic approach

A restrained dominating set [Formula: see text] is said to be a strong restrained dominating set of G if every vertex [Formula: see text] is adjacent to a vertex [Formula: see text] and [Formula: see text]. The minimum cardinality of a strong restrained dominating set of G is said to be the strong restrained domination number of [Formula: see text] and it is denoted by [Formula: see text]. We characterize all the trees with [Formula: see text]. We show that if [Formula: see text] is a tree of order [Formula: see text], then [Formula: see text] and characterize the extremal trees by achieving this lower bound. Simulation was carried out for the strong restrained domination number of Cartesian, strong and lexicographic product under four different network topologies and it was noted that [Formula: see text] under these four network topologies. Furthermore, the strong restrained domination number of [Formula: see text] and [Formula: see text] for any [Formula: see text] and [Formula: see text] for any [Formula: see text] is determined.

Extremal trees for the Randić index

Abstract Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R ( G ) = ∑ x y ∈ Y ( G ) 1 τ ( x ) τ ( y ) R\left( G \right) = \sum\nolimits_{xy \in Y\left( G \right)} {{1 \over {\sqrt {\tau \left( x \right)\tau \left( y \right)} }}} , where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs.