The spectral radius of graphs without trees of diameter at most four

Hou, Xinmin; Liu, Boyuan; Wang, Shicheng; Gao, Jun; Lv, Chenhui

doi:10.1080/03081087.2019.1628911

Cited by 12 publications

(6 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the recent ten years, there are fruitful results of Brualdi-Solheid-Turán type problems, for example, in [2,4,9,11,13,14,16,17,18,19,21].…”

Section: Introductionmentioning

confidence: 99%

Spectral extrema of graphs with bounded clique number and matching number

Wang¹,

Hou²,

Ma³

2023

Preprint

View full text Add to dashboard Cite

For a set of graphs F , let ex(n, F ) and spex(n, F ) denote the maximum number of edges and the maximum spectral radius of an n-vertex F -free graph, respectively. Nikiforov (LAA, 2007) gave the spectral version of the Turán Theorem by showing that spex(n, K k+1 ) = λ(T k (n)), where T k (n) is the k-partite Turán graph on n vertices. In the same year, Feng, Yu and Zhang (LAA) determined the exact value of spex(n, M s+1 ), where M s+1 is a matching with s+1 edges. Recently, Alon and Frankl (arXiv2210.15076) gave the exact value of ex(n, {K k+1 , M s+1 }). In this article, we give the spectral version of the result of Alon and Frankl by determining the exact value of spex(n, {K k+1 , M s+1 }) when n is large.

show abstract

“…In the recent ten years, there are fruitful results of Brualdi-Solheid-Turán type problems, for example, in [2,4,9,11,13,14,16,17,18,19,21].…”

Section: Introductionmentioning

confidence: 99%

Spectral extrema of graphs with bounded clique number and matching number

Wang¹,

Hou²,

Ma³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Conjecture 3 (Nikiforov [27]). Let k 2 and G be a graph of sufficiently large order n. There are many results involving Conjecture 3, see [21,22,27]. Very recently, Conjecture 3 was completely resolved by Cioabǎ, Desai and Tait [7].…”

Section: Introductionmentioning

confidence: 99%

An Aα-Spectral Erdős-Sós Theorem

Chen,

Li,

et al. 2023

Electron. J. Combin.

View full text Add to dashboard Cite

Let $G$ be a graph and let $\alpha$ be a real number in $[0,1].$ In 2017, Nikiforov proposed the $A_\alpha$-matrix for $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_\alpha$-index of $G.$ The famous Erdős-Sós conjecture states that every $n$-vertex graph with more than $\frac{1}{2}(k-1)n$ edges must contain every tree on $k+1$ vertices. In this paper, we consider an $A_\alpha$-spectral version of this conjecture. For $n>k,$ let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and denote by $S^+_{n,k}$ the graph obtained from $S_{n,k}$ by adding one edge. We show that for fixed $k\geq2,\,0<\alpha<1$ and $n\geq\frac{88k^2(k+1)^2}{\alpha^4(1-\alpha)}$, if a graph on $n$ vertices has $A_\alpha$-index at least as large as $S_{n,k}$ (resp. $S^+_{n,k}$), then it contains all trees on $2k+2$ (resp. $2k+3$) vertices, or it is isomorphic to $S_{n,k}$ (resp. $S^+_{n,k}$). These extend the results of Cioabă, Desai and Tait (2022), in which they confirmed the adjacency spectral version of the Erdős-Sós conjecture.

show abstract

“…Partial results towards this conjecture were given in [16,18,19] for trees of diameter at most 4 and other special cases. In this paper we prove Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 99%

A spectral Erdős-Sós theorem

Cioabă¹,

Desai²,

Tait³

2022

Preprint

View full text Add to dashboard Cite

The famous Erdős-Sós conjecture states that every graph of average degree more than t − 1 must contain every tree on t + 1 vertices. In this paper, we study a spectral version of this conjecture. For n > k, let S n,k be the join of a clique on k vertices with an independent set of n − k vertices and denote by S + n,k the graph obtained from S n,k by adding one edge. We show that for fixed k ≥ 2 and sufficiently large n, if a graph on n vertices has adjacency spectral radius at least as large as S n,k and is not isomorphic to S n,k , then it contains all trees on 2k + 2 vertices. Similarly, if a sufficiently large graph has spectral radius at least as large as S + n,k , then it either contains all trees on 2k + 3 vertices or is isomorphic to S + n,k . This answers a two-part conjecture of Nikiforov affirmatively.

show abstract

The spectral radius of graphs without trees of diameter at most four

Cited by 12 publications

References 6 publications

Spectral extrema of graphs with bounded clique number and matching number

Spectral extrema of graphs with bounded clique number and matching number

An Aα-Spectral Erdős-Sós Theorem

A spectral Erdős-Sós theorem

Contact Info

Product

Resources

About