Unicyclic Graphs Research Articles

Ordering Unicyclic Connected Graphs with Girth g ≥ 3 Having Greatest SK Indices

For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees of the vertices. This paper shows a simple and unified approach to the greatest SK indices for unicyclic graphs by using some transformations and characterizes these graphs with the first, second, and third SK indices having order r ≥ 5 and girth g ≥ 3, where girth is the length of the shortest cycle in a graph.

The multiplicity of eigenvalues of unicyclic graphs

Let G be a graph on n vertices. Denote by mG,θ the multiplicity of θ as an eigenvalue of the adjacency matrix of G. In a recent paper, when G is a unicyclic graph, Tian and Wang showed that mG,θ≤⌊n−23⌋ when n≥11 and θ2 is an integer larger than 1. Motivated by it, we establish an upper bound of mG,θ for any fixed real number θ, generalizing the result of Tian and Wang. Some new techniques are developed to realize the generalization.

The Hyper-Zagreb Index of Trees and Unicyclic Graphs

The Hyper-Zagreb Index of Trees and Unicyclic Graphs

Extreme Sombor Spectral Radius of Unicyclic Graphs

Extreme Sombor Spectral Radius of Unicyclic Graphs

Lanzhou Index of Trees and Unicyclic Graphs

Lanzhou Index of Trees and Unicyclic Graphs

Global dominator coloring of unicyclic graphs

Let [Formula: see text]. A vertex [Formula: see text] is a dominator of [Formula: see text] if [Formula: see text] dominates every vertex in [Formula: see text] and [Formula: see text] is said to be an anti-dominator of [Formula: see text] if [Formula: see text] dominates none of the vertices of [Formula: see text]. Let [Formula: see text] be a coloring of [Formula: see text] and let [Formula: see text]. A color class [Formula: see text] is called a dom-color class or an anti-dom-color class of the vertex [Formula: see text] according as [Formula: see text] is a dominator of [Formula: see text] or an anti-dominator of [Formula: see text]. The coloring [Formula: see text] is called a global dominator coloring of [Formula: see text] if every vertex of [Formula: see text] has a dom-color class and an anti-dom-color class in [Formula: see text]. The minimum number of colors required for a global dominator coloring of [Formula: see text] is called the global dominator chromatic number and is denoted by [Formula: see text]. This parameter was introduced in [I. S. Hamid and M. Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 (2019) 325–339]. This paper further extends the study by obtaining the value of [Formula: see text] for unicyclic graphs.

On monophonic position sets in graphs

The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem called the monophonic position problem, obtained by replacing ‘shortest path’ by ‘induced path’. We prove some basic properties and bounds for the monophonic position number of a graph and determine the monophonic position number of some graph families, including unicyclic graphs, complements of bipartite graphs and split graphs. We show that the monophonic position number of triangle-free graphs is bounded above by the independence number. We present realisation results for the general position number, monophonic position number and monophonic hull number. Finally we discuss the complexity of the monophonic position problem.

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 $$P^{*}$$

Minimizing the Gutman Index among Unicyclic Graphs with Given Matching Number

For a connected graph G with vertex set V, denote by d(v) the degree of vertex v and d(u, v) the distance between u and v. The value Gut(G)=∑{u,v}⊆Vd(u)d(v)d(u,v) is called the Gutman index of G. Recently, the graph minimizing the Gutman index among unicyclic graphs with pendent edges was characterized. Denoted by U(n,m) the set of unicyclic graphs on n vertices with matching number m. Motivated by that work, in this paper, we obtain a sharp lower bound for Gutman index of graphs in U(n,m), and the extremal graph attaining the bound is also obtained. It is worth noticing that all the extremal graphs are of high symmetry, that is, they have large automorphic groups.

On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching II

Let G be a simple graph with the adjacency matrix A ( G ) . Let τ ( G ) denote the smallest positive eigenvalue of A ( G ) . In 1990, Pavlíková and Kr c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each m ≥ 3 , let U n be the subclass of U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in U such that τ ( U ) ≤ 2 − 1 . We then find the unique graph U o n (resp. U e n ) having the maximum τ value among all graphs in U n when m is odd (resp. when m is even). Finally, we prove that U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in U . As a consequence, we obtain a sharp upper bound for τ ( U ) when U ∈ U .

TRAVERSING A GRAPH IN GENERAL POSITION

AbstractLetGbe a graph. Assume that to each vertex of a set of vertices$S\subseteq V(G)$a robot is assigned. At each stage one robot can move to a neighbouring vertex. ThenSis a mobile general position set ofGif there exists a sequence of moves of the robots such that all the vertices ofGare visited while maintaining the general position property at all times. The mobile general position number ofGis the cardinality of a largest mobile general position set ofG. We give bounds on the mobile general position number and determine exact values for certain common classes of graphs, including block graphs, rooted products, unicyclic graphs, Kneser graphs$K(n,2)$and line graphs of complete graphs.

On the eigenvalues of Laplacian ABC-matrix of graphs

For a simple graph G, the ABC-index is a degree based topological index and is defined as where dv is the degree of the vertex υ in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by where is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G: The eigenvalues of the matrix are called the Laplacian ABC-eigenvalues of G. In the article, we consider the problem of characterization of connected graphs having exactly three distinct Laplacian ABC-eigenvalues. We solve this problem for bipartite graphs, multipartite graphs, unicyclic graphs, regular graphs and prove the non-existence of such graphs with diameter greater than 2. We introduce the concept of trace norm of the matrix called the Laplacian ABC-energy of G. We obtain some upper and lower bounds for the Laplacian ABC-energy and characterize the extremal graphs which attain these bounds.

Total Global Dominator Coloring of Trees and Unicyclic Graphs

A total global dominator coloring of a graph is a proper vertex coloring of with respect to which every vertex in dominates a color class, not containing and does not dominate another color class. The minimum number of colors required in such a coloring of is called the total global dominator chromatic number, denoted by . In this paper, the total global dominator chromatic number of trees and unicyclic graphs are explored.

On alpha -Excellent Graphs

A graph G is alpha -excellent if every vertex of G is contained in some maximum independent set of G. In this paper, we characterize alpha -excellent bipartite graphs, alpha -excellent unicyclic graphs, alpha -excellent simplicial graphs, alpha -excellent chordal graphs, alpha -excellent block graphs, and we show that every generalized Petersen graph is alpha -excellent.

Maximum and minimum values of inverse degree and forgotten indices on the class of all unicyclic graphs

For a connected simple graph G, the inverse degree index and forgotten index are defined as and respectively, where denotes the degree of vertex u in G. In this article, we use some graph transformations and introduce a method for determining the extremal values of inverse degree index and forgotten index on the class of unicyclic graphs.

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.

Some algebraic invariants of the edge ideals of perfect $ [h, d] $-ary trees and some unicyclic graphs

<abstract><p>This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect $ [h, d] $-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.</p></abstract>

Graphs with mixed metric dimension three and related algorithms

<abstract><p>Let $ G = (V, E) $ be a simple connected graph. A vertex $ x\in V(G) $ resolves the elements $ u, v\in E(G)\cup V(G) $ if $ d_G(x, u)\neq d_G(x, v) $. A subset $ S\subseteq V(G) $ is a mixed metric resolving set for $ G $ if every two elements of $ G $ are resolved by some vertex of $ S $. A set of smallest cardinality of mixed metric generator for $ G $ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $ J_{n}\ncong P_{n} $ with $ n\geq 3 $ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph.</p></abstract>

Characterizing Graphs with Nullity n-4

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of G. A unified approach is presented for the characterization of graphs of order n with η(G) = n−4. All known results on trees, unicyclic graphs, bicyclic graphs, graphs with minimum degree 1, and r-partite graphs, for which η(G) = n−4 are shown to be corollaries of a theorem of Chang, Huang and Yeh that characterizes all graphs with nullity n − 4.

New Diagonal Graph Ramsey Numbers of Unicyclic Graphs

Grossman conjectured that $R(G, G) = 2 \cdot |V(G)| - 1$, for all simple connected unicyclic graphs $G$ of odd girth and $|V(G)| \geq 4$. In this note, we prove his conjecture for various classes of $G$ containing a triangle. In addition, new diagonal graph Ramsey numbers are calculated for some classes of simple connected unicyclic graphs of even girth.