Spectral Radius Of Unicyclic Graphs Research Articles

First zagreb spectral radius of unicyclic graphs and trees

First zagreb spectral radius of unicyclic graphs and trees

Extreme Sombor Spectral Radius of Unicyclic Graphs

Extreme Sombor Spectral Radius of Unicyclic Graphs

On large ABC spectral radii of unicyclic graphs

The ABC spectral radius of a graph G is the largest eigenvalue of the ABC matrix of G, whose (u,v)-entry is du+dv−2dudv if uv∈E(G) and 0 otherwise for u,v∈V(G), where du denotes the degree of u in G, and V(G) (E(G), respectively) denotes the vertex (edge, respectively) set of G. The unicyclic graphs with the first four largest ABC spectral radii are determined.

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.

Distance spectral radius of unicyclic graphs with fixed maximum degree

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

Spectral radius and extremal graphs for class of unicyclic graph with pendant vertices

In this article, we research on the spectral radius of extremal graphs for the unicyclic graphs with girth g mainly by the graft transformation and matching and obtain the upper bounds of the spectral radius of unicyclic graphs.

On the signless Laplacian spectral radius of unicyclic graphs with fixed matching number

We determine the graph with the largest signless Laplacian spectral radius among all unicyclic graphs with fixed matching number.

On the distance spectral radius of unicyclic graphs with perfect matchings

For a connected graph, the distance spectral radius is the largest eigenvalue of its 2k) is the unique graph with the maximum (resp., minimum) distance spectral radius among all unicyclic graphs with perfect matchings on 2k(k > 5) vertices.

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let Un,g be the lollipop graph obtained by appending a pendent vertex of a path on n − g (n > g) vertices to a vertex of a cycle on g ⩾ 3 vertices. We prove that the graph Un,g uniquely minimizes the Laplacian spectral radius for n ⩾ 2g − 1 when g is even and for n ⩾ 3g − 1 when g is odd.

The Largest Laplacian Spectral Radius of Unicyclic Graphs with Fixed Diameter

We identify graphs with the maximal Laplacian spectral radius among all unicyclic graphs withnvertices and diameterd.

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.

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n, d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n, d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].

The Laplacian spectral radius for unicyclic graphs with given independence number

In this paper, we give a complete characterization of the extremal graphs with maximal Laplacian spectral radius among all unicyclic graphs with given order and given number of pendent vertices. Then we study the Laplacian spectral radius of unicyclic graphs with given independence number and characterize the extremal graphs completely.

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 Signless Laplacian Spectral Radius of Unicyclic Graphs with Graph Constraints

In this paper, we study the signless Laplacian spectral radius of unicyclic graphs with prescribed number of pendant vertices or independence number. We also characterize the extremal graphs completely.

Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.

On the spectral radii of unicyclic graphs with fixed matching number

In this paper, the first five sharp upper bounds on the spectral radii of unicyclic graphs with fixed matching number are presented. The first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings are also given, respectively.

New upper bounds on the spectral radius of unicyclic graphs

Let G = ( V ( G ) , E ( G ) ) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C r be the unique cycle of G . The graph G - E ( C r ) is a forest of r rooted trees T 1 , T 2 , … , T r with root vertices v 1 , v 2 , … , v r , respectively. Let k ( G ) = max 1 ⩽ i ⩽ r { max { dist ( v i , u ) : u ∈ V ( T i ) } } + 1 , where dist ( v , u ) is the distance from v to u. Let μ 1 ( G ) and λ 1 ( G ) be the spectral radius of the Laplacian matrix and adjacency matrix of G , respectively. We prove that μ 1 ( G ) < Δ + 2 Δ - 1 cos π 2 k ( G ) + 1 , whenever Δ > 2 and λ 1 ( G ) < 2 Δ - 1 cos π 2 k ( G ) + 1 , whenever Δ ⩾ 4 or whenever Δ = 3 and k ( G ) ⩾ 4 .

On the spectral radius of unicyclic graphs with fixed diameter

In the paper, we will determine graphs with the maximal spectral radius among all the unicyclic graphs with n vertices and diameter d .