Unicyclic Graphs Research Articles

Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is ΩYα(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)+dΩ(ϱ)dΩ(ϱ))α, while the second form of the GPSCI is defined as Yβ(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)×dΩ(ϱ)dΩ(ϱ))β. In this paper, we investigate Yβ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Yβ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.

On the Exponential Atom-Bond Connectivity Index of Graphs

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: eABC=eABC(Υ)=∑vivj∈E(Υ)edi+dj−2didj, where di is the degree of the vertex vi in Υ. In this paper, we prove that the double star DSn−3,1 is the second maximal graph with respect to the eABC index of trees of order n. We give an upper bound on eABC of unicyclic graphs of order n and characterize the maximal graphs. The graph K1∨(P3∪(n−4)K1) gives the maximal graph with respect to the eABC index of bicyclic graphs of order n. We present several relations between eABC(Υ) and ABC(Υ) of graph Υ. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.

Lower bounds on trees and unicyclic graphs with respect to the misbalance rodeg index.

The Misbalance Rodeg (MR) index stands out among the 148 discrete Adriatic indices demonstrating considerable predictive capabilities in evaluations carried out by the International Academy of Mathematical Chemistry. This index excels particularly in forecasting both the enthalpy and the standard enthalpy of vaporization for octane isomers. Despite its significant chemical applicability, the MR index has not been extensively explored in the literature. One objective of this study is to highlight the importance of this graph invariant to the mathematical chemistry community by examining various mathematical properties associated with it. Our investigation specifically aims to ascertain the minimal values of the MR index for all trees and unicyclic graphs with a given order and maximum vertex degree. Additionally, we extend our analysis to molecular trees and provide a characterization of the respective minimal trees and unicyclic graphs.

Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The signless Laplacian matrix of a graph G is defined as Q(G)=D(G)+A(G), where D(G) is the degree-diagonal matrix of G and A(G) is the adjacency matrix of G. The multiplicity of an eigenvalue μ of Q(G) is denoted by mQ(G,μ). Let G=C(T1,…,Tg) be a unicyclic graph with a perfect matching, where Ti is a rooted tree attached at the vertex vi of the cycle Cg. It is proved that Q(G) has 2 as an eigenvalue if and only if g+t is divisible by 4, where t is the number of Ti of even orders. Another main result of this article gives a characterization for a connected graph G with a perfect matching such that mQ(G,2)=θ(G)+1, where θ(G)=|E(G)|−|V(G)|+1 is the cyclomatic number of G.

On the multiplicative sum Zagreb index of molecular graphs

Abstract Multiplicative sum Zagreb index is a modified version of the famous Zagreb indices. For a graph G G , the multiplicative sum Zagreb index is defined as Π 1 * ( G ) = ∏ u v ∈ E ( G ) ( d G ( u ) + d G ( v ) ) {\Pi }_{1}^{* }\left(G)={\prod }_{uv\in E\left(G)}\left({d}_{G}\left(u)+{d}_{G}\left(v)) , where E ( G ) E\left(G) is the edge set of G G and d G ( u ) {d}_{G}\left(u) stands for the degree of vertex u u in G G . In this article, we determine the extremal multiplicative sum Zagreb indices among all n n -vertex molecular trees, molecular unicyclic graphs, molecular bicyclic graphs and molecular tricyclic graphs.

Fibonacci and telephone numbers in extremal unicyclic graphs

In this paper we will show applications of the Fibonacci numbers and the telephone numbers in edge coloured unicyclic graphs. In particular, we give the upper bound for the ( A , 2 B ) -index in unicyclic graphs. Moreover, we give characterization of the extremal graph achieving this maximum value. We also show connections between the number of ( A , 2 B ) -edge colourings in the class of unicyclic graphs and the telephone numbers as well as the Fibonacci numbers.

Degree-based function index of trees and unicyclic graphs

Degree-based function index of trees and unicyclic graphs

Optimization of General Power-Sum Connectivity Index in Uni-Cyclic Graphs, Bi-Cyclic Graphs and Trees by Means of Operations

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures.

On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices

Consider a unicyclic graph G with edge set E(G). Let f be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as If(G)=∑xy∈E(G)f(dG(x),dG(y)), where dG(x) denotes the degree a vertex x in G. This paper determines optimal bounds for If(G) in terms of the order of G and a parameter z, where z is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function f must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices.

When (signless) Laplacian coefficients meet matchings of subdivision

Let G be a graph, whose subdivision is denoted by S(G). Let ϕL(G,x) be the characteristic polynomial of the Laplacian matrix of G. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of ϕL(G,x), in terms of the spanning forests of G. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of S(G), generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.

Bounds on the edge version of the Nirmala index

The edge version of the Nirmala index is defined in terms of edge degrees as [Formula: see text], where [Formula: see text] denotes the degree of the edge [Formula: see text] in graph [Formula: see text] and [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are adjacent. In this paper, we obtain some sharp bounds for the edge version of the Nirmala index in terms of the edge version of topological indices. Also, we give the extremal unicyclic graphs of order [Formula: see text] with the minimum edge version of the Nirmala index.

Well-indumatched Pseudoforests

A graph is called well-indumatched if all of its maximal induced matchings have the same size. Akbari et al. [1] provided a characterization of well-indumatched trees and some results on well-indumatched unicyclic graphs. In this paper, we extend this result by providing a complete structural characterization of well-indumatched pseudotrees, in which each component has no cycle or a unique cycle, by identifying the well-indumatched graph families.

Fractional Vertex Cover Reliability of Graphs

Let G be a graph and let 0 ≤ p , q and p + q ≤ 1 . Suppose that each vertex of G gets a weight of 1 with probability p , 1 2 with probability q , and 0 with probability 1 − p − q , and vertex weight probabilities are independent. The \textit{fractional vertex cover reliability} of G , denoted by FRel ( G ; p , q ) , is the probability that the sum of weights at the end-vertices of every edge in G is at least 1 . In this article, we first provide various computational formulas for FRel ( G ; p , q ) considering general graphs, basic graphs, and graph operations. Secondly, we determine the graphs which maximize FRel ( G ; p , q ) for all values of p and q in the classes of trees, connected unicyclic and bicyclic graphs with fixed order, and determine the graphs which minimize it in the classes of trees and connected unicyclic graphs with fixed order. Our results on optimal graphs extend some known results in the literature about independent sets, and the tools we developed in this paper have the potential to solve the optimality problem in other classes of graphs as well.

Decomposition of the λ -Fold Complete Equipartite Graph into Unicyclic Graphs of Order Five

For a graph G and a subgraph H of a graph G , an H -decomposition of the graph G is a partition of the edge set of G into subsets E i , 1 ≤ i ≤ k , such that each E i induces a graph isomorphic to H . In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the λ -fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

Bipartite unicyclic graphs with a unique perfect matching having the smallest positive eigenvalue equal to

The smallest positive eigenvalue τ ( G ) of a simple graph G is the smallest positive eigenvalue of its adjacency matrix A ( G ) . In [F. J. Zhang and A. Chang, Acyclic molecules with greatest HOMO-LUMO separation, Discrete Applied Mathematics, 98:165–171, (1999).], the authors characterized all nonsingular trees with τ equal to 2 − 1 . We consider the same problem for bipartite unicyclic graphs with a unique perfect matching. Let U be the class of all connected bipartite unicyclic graphs with a unique perfect matching. In this article, we characterize all graphs U in U with the property that τ ( U ) = 2 − 1 . Further, we show that the largest limit point of the smallest positive eigenvalues of graphs in U is 2 − 1 , whereas the smallest limit point is 0.

On the zero forcing number of the complement of graphs with forbidden subgraphs

Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either n−3 or n−1 in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.

Paired coalition in graphs

A paired coalition in a graph G = ( V , E ) consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a paired dominating set but whose union V 1 ∪ V 2 is a paired dominating set. A paired coalition partition (abbreviated pc-partition) in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that each set V i of π is not a paired dominating set but forms a paired coalition with another set V j ∈ π . The paired coalition graph PCG ( G , π ) of the graph G with the pc-partition π of G, is the graph whose vertices correspond to the sets of π , and two vertices V i and V j are adjacent in PCG ( G , π ) if and only if their corresponding sets V i and V j form a paired coalition in G. In this paper, we initiate the study of paired coalition partitions and paired coalition graphs. In particular, we determine the paired coalition number of paths and cycles, obtain some results on paired coalition partitions in trees and characterize pair coalition graphs of paths, cycles and trees. We also characterize triangle-free graphs G of order n with PC ( G ) = n and unicyclic graphs G of order n with PC ( G ) = n − 2 .

Laplacian eigenvalue distribution for unicyclic graphs

Let G be a graph and let mG[0,1) denote the number of Laplacian eigenvalues of G in the interval [0,1). For a tree T with diameter d, Guo, Xue, and Liu proved that mT[0,1)≥(d+1)/3. In this paper, we provide a lower bound for mG[0,1) when G is a unicyclic graph, in terms of the diameter and girth of G. Moreover, for the lollipop graph, under certain conditions on its diameter and girth, we give a formula for the exact value of mG[0,1).

First zagreb spectral radius of unicyclic graphs and trees

First zagreb spectral radius of unicyclic graphs and trees

An improved upper bound for the online graph exploration problem on unicyclic graphs

The online graph exploration problem, which was proposed by Kalyanasundaram and Pruhs (Theor Comput Sci 130(1):125–138, 1994), is defined as follows: Given an edge-weighted undirected connected graph and a specified vertex (called the origin), the task of an algorithm is to compute a path from the origin to the origin which contains all the vertices of the given graph. The goal of the problem is to find such a path of minimum weight. At each time, an online algorithm knows only the weights of edges each of which consists of visited vertices or vertices adjacent to visited vertices. Fritsch (Inform Process Lett 168:1006096, 2021) showed that the competitive ratio of an online algorithm is at most three for any unicyclic graph. On the other hand, Brandt et al. (Theor Comput Sci 839:176–185, 2020) showed a lower bound of two on the competitive ratio for any unicyclic graph. In this paper, we showed the competitive ratio of an online algorithm is at most 5/2 for any unicyclic graph.