Unicyclic Graphs Research Articles

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.

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.

On Sombor Index of Unicyclic and Bicyclic Graphs

Gutman proposed a topological index called the Sombor index, which was defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] in graph [Formula: see text]. In this paper, we determine the second-minimum and second-maximum values of the Sombor index over all the unicyclic graphs of order [Formula: see text] and bicyclic graphs of order [Formula: see text].

On unicyclic graphs with a given girth and their minimum symmetric division deg index

On unicyclic graphs with a given girth and their minimum symmetric division deg index

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

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.

Super Edge-magic Total Strength of Some Unicyclic Graphs

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

Induced dominating sequence and ESD graphs

A vertex subset D of a graph G = (V,E) is said to be a dominating set if every vertex in G is either in D or adjacent to some vertex in D. The minimum cardinality of such a set is the domination number, which is denoted as γ(G). In this paper, we define a sequence associated with the domination concept in graphs and studied the basic properties of the sequence in terms of various parameters of graphs. Using this sequence we order the vertices of a dominating set according its significance and propose Equally Significant Dominating (ESD) graphs. We also introduced domination related topological indices and compute their lower bounds for trees, unicyclic graphs and bicyclic graphs. All the graphs attaining the bounds are characterized.

The second smallest normalized Laplacian eigenvalue of unicyclic graphs

The second smallest normalized Laplacian eigenvalue of unicyclic graphs

Trees, unicyclic graphs and their geometric Sombor index: an extremal approach

Trees, unicyclic graphs and their geometric Sombor index: an extremal approach

Unified approach for spectral properties of weighted adjacency matrices for graphs with degree-based edge-weights

Let G be a graph and di be the degree of a vertex vi in G. For a symmetric real function f(x,y), one can get an edge-weighted graph in such a way that for each edge vivj of G, the weight of vivj is assigned by f(di,dj). Hence, we have a weighted adjacency matrix Af(G) of G, in which the ij-entry is equal to f(di,dj) if vivj∈E(G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of Af(G) for f(x,y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights.

On the unicyclic graphs having vertices that belong to all their (strong) metric bases

A metric basis in a graph G is a smallest possible set S of vertices of G, with the property that any two vertices of G are uniquely recognized by using a vector of distances to the vertices in S. A strong metric basis is a variant of metric basis that represents a smallest possible set S′ of vertices of G such that any two vertices x,y of G are uniquely recognized by a vertex v∈S′ by using either a shortest x−v path that contains y, or a shortest y−v path that contains x. Given a graph G, there exist sometimes some vertices of G such that they forcedly belong to every metric basis or to every strong metric basis of G. Such vertices are called (resp. strong) basis forced vertices in G. It is natural to consider finding them, in order to find a (strong) metric basis in a graph. However, deciding about the existence of these vertices in arbitrary graphs is in general an NP-hard problem, which makes desirable the problem of searching for (strong) basis forced vertices in special graph classes. This article centres the attention in the class of unicyclic graphs. It is known that a unicyclic graph can have at most two basis forced vertices. In this sense, several results aimed to classify the unicyclic graphs according to the number of basis forced vertices they have are given in this work. On the other hand, with respect to the strong metric bases, it is proved in this work that unicyclic graphs can have as many strong basis forced vertices as we would require. Moreover, some characterizations of the unicyclic graphs concerning the existence or not of such vertices are given in the exposition as well.

Regularity of the edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.