Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

Zhai, Mingqing; Lin, Huiqiu; Shu, Jinlong

doi:10.1016/j.ejc.2021.103322

Cited by 52 publications

(31 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Very recently, Nikiforov [17] proved that, if m ≥ (12r) 4 and λ(G) ≥ √ m, then G contains a copy of B r+1 , unless G is a complete bipartite graph (possibly with some isolated vertices). This result further extends above two theorems and solves a conjecture proposed in [22].…”

Section: Theorem 2 ([22]supporting

confidence: 89%

See 1 more Smart Citation

Counting substructures and eigenvalues II: quadrilaterals

Niu¹,

Zhai²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let G be a graph and λ(G) be the spectral radius of G. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph G on m ≥ 10 edges contains a 4to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that f (m) = Ω(m). In this article, by somewhat different techniques, we prove that f (m) = Θ(m 2 ). We left the solution to lim m→∞ f (m) m 2 as a problem, and also mention other ones for further study.

show abstract

Section: Theorem 2 ([22]supporting

confidence: 89%

“…unless it is a complete bipartite graph with possibly some isolated vertices (see [17], this confirmed a conjecture in [22]).…”

Section: Nikiforov's Deleting Small Eigenvalue Edge Methodssupporting

confidence: 54%

Counting substructures and eigenvalues II: quadrilaterals

Niu¹,

Zhai²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…During the years this striking and elegant result has attracted significant attention (see, e.g., [8] and its references for some highlights.) In this note we discuss two recent developments of Nosal's result.…”

Section: Introductionmentioning

confidence: 96%

On a theorem of Nosal

Nikiforov¹

2021

Preprint

View full text Add to dashboard Cite

Let G be a graph with m edges and spectral radius λ 1 . Let bk (G) stand for the maximal number of triangles with a common edge in G.In 1970 Nosal proved that if λ 2 1 > m, then G contains a triangle. In this paper we show that the same premise implies thatThis result settles a conjecture of Zhai, Lin, and Shu. Write λ 2 for the second largest eigenvalue of G. Recently, Lin, Ning, and Wu showed that if G is a triangle-free graph of order at least three, thenthereby settling the simplest case of a conjecture of Bollobás and the author. We give a simpler proof of their result.

show abstract

“…Notice that Nikiforov [30] characterized the graph among the set of n-vertex C 3 -free graphs having the maximum spectral radius; he also characterized the graph among the set of n-vertex C 4 -free graphs having the maximum spectral radius (see [31]). Very recently, Zhai, Lin and Shu [45] characterized the graph among the set of n-vertex C 5 -free (or C 6 -free) graphs having the maximum spectral radius. We summarize them as follows.…”

Section: Further Discussionmentioning

confidence: 99%

“…, then G contains C t for every t 6 unless G ∼ = S m+3 2 ,2 . Inspired by Theorem 5.1, Zhai, Lin and Shu [45] proposed the following conjecture (see, [45,Conjecture 5.1]), which gives a more general spectral characterization of graphs containing cycles with consecutive lengths. Conjecture 5.2.…”

Section: Further Discussionmentioning

confidence: 99%

Adjacency eigenvalues of graphs without short odd cycles

Li¹,

Sun²,

Yu³

2021

Preprint

View full text Add to dashboard Cite

It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ 1 and λ 2 the largest and the second largest eigenvalues of the adjacency matrix A(G) of G, respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper boundwhere k is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most 2k + 1 in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of n-vertex non-bipartite graphs with odd girth at least 2k + 3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

show abstract

Spectral extrema of graphs with fixed size: Cycles and complete bipartite graphs

Cited by 52 publications

References 18 publications

Counting substructures and eigenvalues II: quadrilaterals

Counting substructures and eigenvalues II: quadrilaterals

On a theorem of Nosal

Adjacency eigenvalues of graphs without short odd cycles

Contact Info

Product

Resources

About