The minimum spectral radius of graphs with a given independence number

Xu, Mimi; Yan, Hong; Shu, Jinlong; Zhai, Mingqing

doi:10.1016/j.laa.2009.03.055

Cited by 27 publications

(22 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…However, a proof by induction on α gives the following sharper result. In a different direction, for connected graphs and some special values of α, more specific results have been proved in [91].…”

Section: Spectral Radius and Independence Numbermentioning

confidence: 99%

Some new results in extremal graph theory

Nikiforov¹

2011

Surveys in Combinatorics 2011

View full text Add to dashboard Cite

In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simplified and streamlined. These results include a new Erdős-Stone-Bollobás theorem, several stability theorems, several saturation results and bounds for the number of graphs with large forbidden subgraphs.Another recent trend is the expansion of spectral extremal graph theory, in which extremal properties of graphs are studied by means of eigenvalues of various matrices. One particular achievement in this area is the casting of the central results above in spectral terms, often with additional enhancement. In addition, new, specific spectral results were found that have no conventional analogs.All of the above material is scattered throughout various journals, and since it may be of some interest, the purpose of this survey is to present the best of these results in a uniform, structured setting, together with some discussions of the underpinning ideas.

show abstract

Section: Spectral Radius and Independence Numbermentioning

confidence: 99%

Some new results in extremal graph theory

Nikiforov¹

2011

Surveys in Combinatorics 2011

View full text Add to dashboard Cite

show abstract

“…In Reference [6], the extremal graph with maximal α-index with fixed order and cut vertices and the extremal tree which attains the maximal α-index with fixed order and matching number are characterized. In Reference [7], the authors obtain the extremal graphs with maximal α-index with fixed order and diameter at least k. In References [8,9], are characterized the graphs which have the minimal spectral radius among all the connected graphs of order n and some values of the independence number γ. In Reference [10], Nikiforov et al shown that if T ∆ is a tree of maximal degree ∆, then the spectral radius of A α (T ∆ ) satisfies the tight inequality…”

Section: Its Straightforward Verified Thatmentioning

confidence: 99%

New Bounds for the α-Indices of Graphs

2020

View full text Add to dashboard Cite

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

show abstract

“…However, the other end of the problem turns out to be hard. Recently, Xu et al [15] determined the connected graphs of order n with independence number α ∈ {1, 2, n 2 , n 2 + 1, n − 3, n − 2, n − 1} which minimize the spectral radius. We call such graphs minimal graphs.…”

Section: Introductionmentioning

confidence: 99%

Graphs With Small Independence Number Minimizing the Spectral Radius

Shi

2013

Discrete Math. Algorithm. Appl.

View full text Add to dashboard Cite

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937-945] determined the connected graphs of order n with independence number α ∈ {1, 2, n 2 , n 2 + 1, n − 3, n − 2, n − 1} which minimize the spectral radius. In this paper, we show that the graph obtained from a path of order α by blowing up each vertex to a clique of order k minimizes the spectral radius among all connected graphs of order kα with independence number α for α = 3, 4 and conjecture that this is true for all α ∈ N.

show abstract

The minimum spectral radius of graphs with a given independence number

Cited by 27 publications

References 9 publications

Some new results in extremal graph theory

Some new results in extremal graph theory

New Bounds for the α-Indices of Graphs

Graphs With Small Independence Number Minimizing the Spectral Radius

Contact Info

Product

Resources

About