Unicyclic Graphs Research Articles

Vertex and edge metric dimensions of cacti

In a graph G , a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S . The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G . In Sedlar and Škrekovski (0000) we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general.

A linear-time algorithm for minimum k-hop dominating set of a cactus graph

Given a graph G = ( V , E ) and an integer k ≥ 1 , a k - hop dominating set D of G is a subset of V , such that, for every vertex v ∈ V , there exists a node u ∈ D whose distance from v is at most k . A k -hop dominating set of minimum cardinality is called a minimum k -hop dominating set . In this paper, we present a linear-time algorithm that finds a minimum k -hop dominating set in cactus graphs , which improves the O ( n 3 ) -time algorithm of Borradaile and Le (2017). To achieve this, we show that the k -hop dominating set problem for unicyclic graphs reduces to the piercing circular arcs problem , and show a linear-time algorithm for piercing sorted circular arcs, which improves the best known O ( n log n ) -time algorithm.

On General Reduced Second Zagreb Index of Graphs

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRMα, known under the name general reduced second Zagreb index, is defined as GRMα(Γ)=∑uv∈E(Γ)(dΓ(u)+α)(dΓ(v)+α), where dΓ(v) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to GRMα(α≥−12). Using the extremal unicyclic graphs, we obtain a lower bound on GRMα(Γ) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on GRMα of different classes of graphs in terms of order n, size m, independence number γ, chromatic number k, etc. In particular, we present an upper bound on GRMα of connected triangle-free graph of order n>2, m>0 edges with α>−1.5, and characterize the extremal graphs. Finally, we prove that the Turán graph Tn(k) gives the maximum GRMα(α≥−1) among all graphs of order n with chromatic number k.

Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Given a simple connected graph G, the metric dimension dim(G) (and edge metric dimension edim(G)) is defined as the cardinality of a smallest vertex subset S⊆V(G) for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S. It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim(G) and edim(G) are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.

The effect of graph operations on the degree-based entropy

The degree-based entropy Id(G) of a graph G on m>0 edges is obtained from the well-known Shannon entropy −∑i=1np(xi)logp(xi) in information theory by replacing the probabilities p(xi) by the fractions dG(vi)2m, where {v1,v2,…,vn} is the vertex set of G, and dG(vi) is the degree of vi. We continue earlier work on Id(G). Our main results deal with the effect of a number of graph operations on the value of Id(G). We also illustrate the relevance of these results by applying some of these operations to prove a number of extremal results for the degree-based entropy of trees and unicyclic graphs.

Unicyclic graphs with extremal Lanzhou index

Unicyclic graphs with extremal Lanzhou index

Spectra of closeness Laplacian and closeness signless Laplacian of graphs

For a graph G with vertex set V(G) and u, v ∈ V(G), the distance between vertices u and v in G, denoted by dG(u,v), is the length of a shortest path connecting them and it is ∞ if there is no such a path, and the closeness of vertex u in G is cG(u) = ∑w∈V(G)2-dG(u,w). Given a graph G that is not necessarily connected, for u, v∈V(G), the closeness matrix of G is the matrix whose (u,v)-entry is equal to 2-dG(u,v) if u≠v and 0 otherwise, the closeness Laplacian is the matrix whose (u,v)-entry is equal to $$ \left\{\begin{array}{c}-{2}^{-{d}_G(u,v)}\hspace{1em}\mathrm{if}\enspace u\ne v,\enspace \\ \enspace {c}_G(u)\hspace{1em}\hspace{1em}\mathrm{otherwise}\hspace{0.5em}\end{array}\right.\hspace{0.5em} $$ and the closeness signless Laplacian is the matrix whose (u,v)-entry is equal to $$ \left\{\begin{array}{c}{2}^{-{d}_G(u,v)}\hspace{1em}\hspace{1em}\&\mathrm{if}\enspace \mathrm{u}\ne \mathrm{v},\\ {c}_G(u)\hspace{1em}\hspace{1em}\mathrm{otherwise}.\end{array}\right. $$ We establish relations connecting the spectral properties of closeness Laplacian and closeness signless Laplacian and the structural properties of graphs. We give tight upper bounds for all nontrivial closeness Laplacian eigenvalues and characterize the extremal graphs, and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian eigenvalue. Also, we give tight upper bounds for the closeness signless Laplacian eigenvalues and determine the trees whose largest closeness signless Laplacian eigenvalues achieve the first two largest values.

The extremal Sombor index of trees and unicyclic graphs with given matching number

In 2021, the Sombor index was introduced by Gutman, which is a new vertex-degree-based topological index. The Sombor index of a graph G is defined as , where d G (υ) is the degree of the vertex υ in G. Let and be the set of trees and unicyclic graphs on n vertices with fixed matching number m, respectively. In this paper, the trees and the unicyclic graphs with the maximum Sombor index among and are characterized, respectively.

Combinatorial explanation of the weighted Wiener (Kirchhoff) index of trees and unicyclic graphs

Let G be a vertex-weighted graph with vertex-weighted function ω:V(G)→R+ satisfying ω(vi)=xi for each vi∈V(G)={v1,v2,…,vn}. The weighted Wiener index of vertex-weighted graph G is defined as W(G;x1,x2,…,xn)=∑1≤i<j≤nxixjdG(vi,vj), where dG(vi,vj) denotes the distance between vi and vj in G. In this paper, we give a combinatorial explanation of W(G;x1,x2,…,xn) when G is a tree: W(G;x1,x2,…,xn)=m(S(G),n−2)∏i=1nxi, where m(S(G),n−2) is the sum of weights of matchings of some weighted subdivision S(G) of G with n−2 edges. We also give a similar combinatorial explanation of the weighted Kirchhoff index K(G;x1,x2,…,xn)=∑1≤i<j≤nxixjrG(vi,vj) of a unicyclic graph G, where rG(vi,vj) denotes the resistance distance between vi and vj in G. These results generalize some known formulae on the Wiener and Kirchhoff indices.

Survey on the general Randić index: extremal results and bounds

For α∈ℝ, the general Randić index of a graph G is defined as Rα(G)=∑uv∈E(G)[d(u)d(v)]α, where E(G) is the edge set of G, and d(u) and d(v) are the degrees of vertices u and v in G, respectively. We give a survey containing known extremal results about the general Randić index. We present results for general graphs, trees, unicyclic graphs, bicyclic graphs and chemical graphs.

The k-Szeged index of graphs

Based on (revised) Szeged index of a graph and the Steiner k-Wiener index of a tree, we introduce the k-Szeged index Szk(G) and revised k-Szeged index Szk⁎(G) of a graph G=(V,E), defined as Szk⁎(G)=∑e=uv∈E(G)∑i=1k−1(nu(e)+n0(e)2i)(nv(e)+n0(e)2k−i) and Szk(G)=∑e=uv∈E(G)∑i=1k−1(nu(e)i)(nv(e)k−i), where nu(e), nv(e) and n0(e) denote respectively the number of vertices of G lying closer to vertex u than to vertex v, the number of vertices of G lying closer to vertex v than to vertex u and the number of vertices with equal distance to u and v. In this paper, we first determine upper and lower bounds of (revised) k-Szeged indices of a connected graph G in terms of the numbers of vertices, edges and pendant edges, and give Nordhaus-Gaddum-type results of (revised) k-Szeged indices. Then we determine the extremal value of these indices and the corresponding extremal graphs among all complete bipartite graphs with n vertices. Finally, we discuss the upper and lower bounds of these indices for unicyclic graphs.

On the multiplicity of positive eigenvalues of a graph

Let T be a tree on n(≥7) vertices with λ as a positive eigenvalue of multiplicity k. If λ2≥2 is an integer, then we prove that k≤⌊n−43⌋ and all extremal graphs attaining the upper bound are characterized. This result revises and improves the main conclusion of Wong, Zhou and Tian (2020). Moreover, applying this result we investigate the eigenvalue multiplicity of unicyclic graphs. Let G be a unicyclic graph of order n(≥11), which contains λ (λ2≥2 is an integer) as a positive eigenvalue of multiplicity m. Then it is proved that m≤⌊n−23⌋, and all extremal graphs attaining the upper bound are determined. These two upper bounds improve the conclusions of Rowlinson (2010, 2011), respectively.

The local metric dimension of split and unicyclic graphs

A set &lt;em&gt;W&lt;/em&gt; is called a local resolving set of &lt;em&gt;G&lt;/em&gt; if the distance of &lt;em&gt;u&lt;/em&gt; and &lt;em&gt;v&lt;/em&gt; to some elements of &lt;em&gt;W&lt;/em&gt; are distinct for every two adjacent vertices &lt;em&gt;u&lt;/em&gt; and &lt;em&gt;v&lt;/em&gt; in &lt;em&gt;G&lt;/em&gt;. The local metric dimension of &lt;em&gt;G&lt;/em&gt; is the minimum cardinality of a local resolving set of &lt;em&gt;G&lt;/em&gt;. A connected graph &lt;em&gt;G&lt;/em&gt; is called a split graph if &lt;em&gt;V&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) can be partitioned into two subsets &lt;em&gt;V&lt;/em&gt;&lt;sub&gt;1&lt;/sub&gt; and &lt;em&gt;V&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt; where an induced subgraph of G by &lt;em&gt;V&lt;/em&gt;&lt;sub&gt;1&lt;/sub&gt; and &lt;em&gt;V&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt; is a complete graph and an independent set, respectively. We also consider a graph, namely the unicyclic graph which is a connected graph containing exactly one cycle. In this paper, we provide a general sharp bounds of local metric dimension of split graph. We also determine an exact value of local metric dimension of any unicyclic graphs.

The odd-valued chromatic polynomial of a signed graph

For a signed graph Σ=(G,σ), Zaslavsky defined a proper coloring on Σ and showed that the function counting the number of such colorings is a quasi-polynomial with period two, that is, a pair of polynomials, one for odd values and the other for even values. In this paper, we focus on the case of odd, written as χ(Σ,x) for short. We initially give a homomorphism expression of such colorings, by which the symmetry is considered in counting the number of homomorphisms. Besides, the explicit formulas χ(Σ,x) for some basic classes of signed graphs are presented. As a main result, we give a combinatorial interpretation of the coefficients in χ(Σ,x) and present several applications. In particular, the constant term in χ(Σ,x) gives a new criterion for balancing and a characterization for unbalanced unicyclic graph. At last, we also give a tight bound for the constant term of χ(Σ,x).

On Spectral Characterization of Two Classes of Unicycle Graphs

Let G be a graph with n vertices, let A(G) be an adjacency matrix of G and let PA(G,λ) be the characteristic polynomial of A(G). The adjacency spectrum of G consists of eigenvalues of A(G). A graph G is said to be determined by its adjacency spectrum (DS for short) if other graphs with the same adjacency spectrum as G are isomorphic to G. In this paper, we investigate the spectral characterization of unicycle graphs with only two vertices of degree three. We use G21(s1,s2) to denote the graph obtained from Q(s1,s2) by identifying its pendant vertex and the vertex of degree two of P3, where Q(s1,s2) is the graph obtained by identifying a vertex of Cs1 and a pendant vertex of Ps2. We use G31(t1,t2) to denote the graph obtained from circle with the vertices v0v1⋯vt1+t2+1 by adding one pendant edge at vertices v0 and vt1+1, respectively. It is shown that G21(s1,s2) (s1≠4,6, s1≥3, s2≥3) and G31(t1,t2) (t1+t2≠2, t2≥t1≥1) are determined by their adjacency spectrum.

Maximum and minimum Sombor index among k-apex unicyclic graphs and k-apex trees

The Sombor index [Formula: see text] of a graph [Formula: see text] is defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] of [Formula: see text]. A [Formula: see text]-cone [Formula: see text]-cyclic graph is the join of the complete graph [Formula: see text] and a connected [Formula: see text]-cyclic graph. A [Formula: see text]-apex tree (respectively, [Formula: see text]-apex unicyclic graph) is defined as a connected graph [Formula: see text] with a [Formula: see text]-subset [Formula: see text] such that [Formula: see text] is a tree (respectively, unicyclic graph), but [Formula: see text] is not a tree (respectively, unicyclic graph) for any [Formula: see text] with [Formula: see text]. In this paper, we show the minimal graphs of [Formula: see text] among all [Formula: see text]-cone [Formula: see text]-cyclic graphs with [Formula: see text] as their degree sequence, and determine the extremal values and extremal graphs of [Formula: see text] among [Formula: see text]-apex unicyclic graphs and [Formula: see text]-apex trees, respectively.

Retraction Note to: An investigation of unicyclic graphs in which the isolate bondage number is equal to three in graph network theory

Retraction Note to: An investigation of unicyclic graphs in which the isolate bondage number is equal to three in graph network theory

Sanskruti Index of some Chemical Trees and Unicyclic Graphs

In chemical graph theory, topological indices are used to estimate the bio-activity of chemical compounds. The molecular graph models molecular compounds. A molecular graph represents the structural formula of a chemical compound relative to the graph. Its vertices represent the atoms of the compound, and its edges represent the chemical bonds. In 2017, Hosamani introduced the Sanskruti index S(G) and defined it as S(G) = Σuv∈E(G) ((δu ⋅ δv)/(δu + δv − 2))3, where δu is the sum of the degrees of all neighbors of the vertex u in G. The Sanskruti index displays a good connection with an entropy of octane isomers. In this paper, the Sanskruti index S(G) is computed for some chemical trees and unicyclic graphs.

Stepwise Irregular Graphs and Their Metric-Based Resolvability Parameters

If the degrees of any two consecutive vertices differ by exactly one, the graph is called a stepwise irregular graph. The study of choosing specific vertices in an ordered subset of the vertex set such that no two vertices have the same representations with regard to the chosen subset is known as resolving parameters. This concept has been expanded for edges as well as a combined version of both. We examine a unique unicyclic stepwise irregular graph and an extended structure of stepwise irregular graph in terms of resolvability parameters to connect the stepwise irregular graph and resolving parameter concepts.

Computing Strong Roman Domination of Trees and Unicyclic Graphs in Linear Time

Computing Strong Roman Domination of Trees and Unicyclic Graphs in Linear Time