On the spectral radius of graphs without a star forest

https://doi.org/10.1016/j.disc.2020.112269Get rights and content

Abstract

Let F=i=1kSdi be the union of pairwise vertex-disjoint k stars of order d1+1,,dk+1, respectively, where k2 and d1dk1. In this paper, we present two sharp upper bounds for the spectral radius of F-free (bipartite) graphs and characterize all corresponding extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of an F-free graph and all extremal graphs are obtained.

Introduction

Let G be an undirected simple graph with vertex set V(G)={v1,,vn} and edge set E(G), where n is called the order of G. The adjacency matrix A(G) of G is the n×n matrix (aij), where aij=1 if vi is adjacent to vj, and 0 otherwise. The spectral radius of G is the largest eigenvalue of A(G), denoted by ρ(G). The least eigenvalue of A(G) is denoted by ρn(G). For vV(G), the degree dG(v) of v is the number of vertices which are adjacent to v in G. We write d(v) for dG(v) if there is no ambiguity. Denote by Δ(G) the maximum degree of G. Let Sn1 be a star of order n. The center of a star is the vertex of maximum degree in the star. A star forest is a forest whose components are stars. The centers of a star forest are the centers of the stars in the star forest. A graph G is H-free if it does not contain H as a subgraph. The classical Turán number, denoted by ex(n,H), is the maximum number of edges in an H-free graph of order n. For X,YV(G), e(X) denotes the number of edges in G with two ends in X, and e(X,Y) denotes the number of edges in G with one end in X and the other in Y. For two vertex disjoint graphs G and H, we denote by GH and GH the union of G and H, and the join of G and H which is obtained from GH by joining each vertex of G to each vertex of H, respectively. Denote by kG the union of k pairwise vertex-disjoint copies of G. For graph notation and terminology undefined here, readers are referred to [1].

One of the fundamental questions in extremal graph theory, which is called Turán problem, is to study the Turán number ex(n,H) in an H-free graph of order n. Among these, determining the Turán number of a star forest has attracted much interest. For example, Lidický, Liu, and Palmer [13] determined the Turán number ex(n,i=1kSdi) of a star forest F=i=1kSdi for sufficiently large n. Yuan and Zhang [21] determined the Turán number ex(n,kS2) of a star forest F=kS2 for all n and characterized all extremal graphs. Recently, Yin and Rao [20] determined the Turán number ex(n,kSl) for all n with k=2,3. Lan, Li, Shi, and Tu [9] further obtained the Turán number ex(n,kSl) for nk(l2+l+1)l2(l3), where k2 and l3. In 2020, Li, Li, and Yin [11] gave the Turán number ex(n,Sd1Sd2) for almost all n. For more results on this topic, readers may be referred to [3], [10], [12], [21].

In spectral extremal graph theory, a similar central problem, which is called the spectral Turán problem, is of the following type (see [16]):

Problem 1.1 [16]

Given a graph H, what is the maximum ρ(G) of any H-free graph G of order n ?

This problem is intensively investigated in the literature for many classes of graphs. For example, Guiduli [8] and Nikiforov [15] independently determined the maximum value of the spectral radius of all Kr-free graph G of order n. Tait [17] gave the maximum value of the spectral radius of all Kr-minor free graphs of order n with r3. For other classes of graphs, the reader may be referred to [2], [4], [7], [18], [22]. In addition, Nikiforov [16] gave an excellent survey on this topic.

In this paper, motivated by Problem 1.1 and the results of Turán number of a star forest, we determine the maximum value of the spectral radius ρ(G) and characterize all corresponding extremal graphs for a star-forest-free (bipartite) graph G of order n. Moreover, we present a sharp lower bound for the minimum least eigenvalue ρn(G) and characterize all extremal graphs for a star-forest-free graph G of order n. In order to state our results, we first introduce some symbols and notations.

Let Fn,k=Kk1(pK2qK1), where n(k1)=2p+q and 0q<2. In addition, for k2 and d1dk1, define f(k,d1,,dk)=(i=1kdi+3k4)2(i=1kdi+k2)2k1+k2,g(k,d1,,dk)=(k1)(i=1kdi+k2)2(i=1kdi+3k1)4k6+i=1k2di+k1.

The main results of this paper can be stated as follows:

Theorem 1.2

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free graph of order n, where nf2(k,d1,,dk)k1+k1, then ρ(G)k+dk3+(kdk1)2+4(k1)(nk+1)2with equality if and only if G=Kk1H, where H is a (dk1)-regular graph of order nk+1.

Theorem 1.3

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free bipartite graph of order n, where ng2(k,d1,,dk)4(k1)+k1, then ρ(G)(k1)(nk+1)with equality if and only if G=Kk1,nk+1.

Corollary 1.4

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free graph of order n, where ng2(k,d1,,dk)4(k1)+k1, then ρn(G)(k1)(nk+1)with equality if and only if G=Kk1,nk+1.

The rest of this paper is organized as follows. In Section 2, some preliminary results and lemmas are presented. In Section 3, we present a proof of Theorem 1.2. In Section 4, we give proofs of Theorem 1.3 and Corollary 1.4.

Section snippets

Preliminary

We first give a very rough estimation on the Turán number ex(n,i=1kSdi) for ni=1kdi+k.

Lemma 2.1

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free graph of order n with ni=1kdi+k, then e(G)12(i=1kdi+2k3)n12(k1)(i=1kdi+k1).

Proof

Let C={vV(G):d(v)i=1kdi+k1}. Since G is F-free, |C|k1 by the definition of C. Hence 2e(G)=vCd(v)+vV(G)Cd(v)(n1)|C|+(n|C|)(i=1kdi+k2)=(ni=1kdik+1)|C|+(i=1kdi+k2)n(k1)(ni=1kdik+1)+(i=1kdi+k2)n=(i=1kdi+2k3)n(k1)(i=

Proof of Theorem 1.2

In this section, we will present the proof of Theorem 1.2 and several corollaries. Before that, we prove the following important result for star-forest-free connected graphs.

Theorem 3.1

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free connected graph of order n, where nf(k,d1,,dk)+1, then ρ(G)k+dk3+(kdk1)2+4(k1)(nk+1)2with equality if and only if G=Kk1H, where H is a (dk1)-regular graph of order nk+1.

Proof

Let G be a connected graph with the maximum spectral radius

Proofs of Theorem 1.3 and Corollary 1.4

In this section, we present the proof of Theorem 1.3 and Corollary 1.4. Before that, we prove the following important result for star-forest-free bipartite connected graphs.

Theorem 4.1

Let F be a star forest, where F=i=1kSdi with k2 and d1dk1. If G is an F-free connected bipartite graph of order n, where ng(k,d1,,dk), then ρ(G)(k1)(nk+1)with equality if and only if G=Kk1,nk+1.

Proof

Let G be a connected graph with the maximum spectral radius among all F-free connected graphs of order n. Set for short

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the anonymous referees for many helpful comments and suggestions.

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    This work is partly supported by the National Natural Science Foundation of China (Nos. 11971311, 12026230, 11531001), the Montenegrin-Chinese Science and Technology Cooperation Project (No. 3-12).

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