Set Of Unicyclic Graphs Research Articles

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.

Extremal values of the Sombor index in unicyclic and bicyclic graphs

Let G be a graph with set of vertices $$V\left( G\right)$$ and set of edges $$E\left( G\right)$$ . The Sombor index is a recently introduced vertex-degree-based-topological index, defined as $$SO(G) = \sum\limits_{uv \in E(G)} {\sqrt {{{({d_u})}^2} + {{({d_v})}^2}} } ,$$ where $$d_{u}$$ denotes the degree of the vertex u. In this paper we study the extremal values of SO over the set of unicyclic graphs and over the set of bicyclic graphs.

General sum-connectivity index of unicyclic graphs with given diameter

For α∈R, the general sum-connectivity index of a graph G is defined as χα(G)=∑uv∈E(G)[degG(u)+degG(v)]α, where E(G) is the edge set of G and degG(v) is the degree of a vertex v in G. Let Un,d be the set of unicyclic graphs with n vertices and diameter d.We present the graph with the smallest general sum-connectivity index among the graphs in Un,d for −1≤α<0 and the graph with the largest general sum-connectivity index among the graphs in Un,d for 0<α<1. Sharp lower bounds on the classical sum-connectivity index and the harmonic index for graphs in Un,d follow from the lower bound on the general sum-connectivity index.

On the Maximal-Adjacency-Spectrum Unicyclic Graphs with Given Maximum Degree

In this paper, we study the properties and structure of the maximal-adjacency-spectrum unicyclic graphs with given maximum degree. We obtain some necessary conditions on the maximal-adjacency-spectrum unicyclic graphs in the set of unicyclic graphs with n vertices and maximum degree Δ and describe the structure of the maximal-adjacency-spectrum unicyclic graphs in the set. Besides, we also give a new upper bound on the adjacency spectral radius of unicyclic graphs, and this new upper bound is the best upper bound expressed by vertices n and maximum degree Δ from now on.

Wiener index of unicycle graphs with given number of even degree vertices

The Wiener index of a connected graph is the sum of the distance of all pairs of distinct vertices. It was introduced by Wiener in 1947 to analyze some aspects of branching by fitting experimental data for several properties of alkane compounds. Denote by [Formula: see text] the set of unicyclic graphs with [Formula: see text] vertices and [Formula: see text] vertices of even degree. In this paper, we present a structural result on the graphs in [Formula: see text] with minimum Wiener index and completely characterize such graphs when [Formula: see text].

Further Results on the Resistance-Harary Index of Unicyclic Graphs

The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.

The signless Laplacian coefficients and the incidence energy of unicyclic graphs with given pendent vertices

Let Urn be the set of unicyclic graphs with n vertices and r pendent vertices (namely, r leaves), where n ≥ 4 and r ≥ 1. We consider the signless Laplacian coefficients (SLCs) and the incidence energy (IE) in Urn. Firstly, among a subset of Urn in which each graph has a fixed odd girth g ≥ 3, where n ≥ g + 1 and r ≥ 1, we characterize a unique extremal graph which has the minimum SLCs and the minimum IE. Secondly, if G  Urn and G has odd girth g ≥ 5, where n ≥ 7 and r ≥ 1, then we prove that a unique extremal graph Ln  Urn with girth 4 satisfies that both the SLCs and the IE of G are more than the counterparts of Ln.

A Note on the Nullity of Unicyclic Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.

The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

SPECTRAL RADII OF UNICYCLIC GRAPHS WITH FIXED NUMBER OF CUT VERTICES

Let [Formula: see text] be the set of unicyclic graphs with n vertices and k cut vertices. In this paper, we determine the unique graph with the maximal spectral radius among all graphs in [Formula: see text] for 1 ≤ k ≤ n - 3.

Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U ( n , d ) be the set of unicyclic graphs on n vertices with diameter d . In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U ( n , d ) . It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449–457].

On the spectral radius of unicyclic graphs with prescribed degree sequence

We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence.

The least eigenvalue of unicyclic graphs with [formula omitted] vertices and [formula omitted] pendant vertices

Let U ( n , k ) be the set of unicyclic graphs with n vertices and k pendant vertices. In this paper, we determine the unique graph with the minimal least eigenvalue among all graphs in U ( n , k ) . The work is related with that of Guo [S.G. Guo, The spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices, Linear Algebra Appl. 408 (2005) 78–85], which determined the unicyclic graph with the maximal spectral radius in U ( n , k ) . We can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius.

Minimizing a class of unicyclic graphs by means of Hosoya index

The Hosoya index, denoted by z ( G ) , of a (molecular) graph G is defined as the total number of independent-edge sets of G . Let U n be the set of unicyclic graphs with n vertices. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. Denote by U n 1 the set of fully loaded unicyclic graphs. In this paper, graphs in U n 1 with minimal, second-minimal and third-minimal Hosoya indices are uniquely determined, respectively.

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let $$S_{n}^{l}$$ be the graph obtained by identifying the center of the star S n-l+1 with any vertex of C l . By $$R_{n}^{l,\,k}$$ we denote the graph obtained by identifying one pendent vertex of the path P n-l-k+1 with one pendent vertex of $$S_{l+k}^{l}$$ . In this paper, we show that $$R_{n}^{l,\,k}$$ is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).