On the spectral radius of graphs without a star forest☆
Introduction
Let be an undirected simple graph with vertex set and edge set , where is called the order of . The adjacency matrix of is the matrix , where if is adjacent to , and otherwise. The spectral radius of is the largest eigenvalue of , denoted by . The least eigenvalue of is denoted by . For , the degree of is the number of vertices which are adjacent to in . We write for if there is no ambiguity. Denote by the maximum degree of . Let be a star of order . 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 is -free if it does not contain as a subgraph. The classical Turán number, denoted by , is the maximum number of edges in an -free graph of order . For , denotes the number of edges in with two ends in , and denotes the number of edges in with one end in and the other in . For two vertex disjoint graphs and , we denote by and the union of and , and the join of and which is obtained from by joining each vertex of to each vertex of , respectively. Denote by the union of pairwise vertex-disjoint copies of . 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 in an -free graph of order . 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 ) of a star forest for sufficiently large . Yuan and Zhang [21] determined the Turán number of a star forest for all and characterized all extremal graphs. Recently, Yin and Rao [20] determined the Turán number for all with . Lan, Li, Shi, and Tu [9] further obtained the Turán number for , where and . In 2020, Li, Li, and Yin [11] gave the Turán number for almost all . 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 , what is the maximum of any -free graph of order
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 -free graph of order . Tait [17] gave the maximum value of the spectral radius of all -minor free graphs of order with . 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 and characterize all corresponding extremal graphs for a star-forest-free (bipartite) graph of order . Moreover, we present a sharp lower bound for the minimum least eigenvalue and characterize all extremal graphs for a star-forest-free graph of order . In order to state our results, we first introduce some symbols and notations.
Let , where and . In addition, for and , define
The main results of this paper can be stated as follows:
Theorem 1.2 Let be a star forest, where with and . If is an -free graph of order , where , then with equality if and only if , where is a -regular graph of order .
Theorem 1.3 Let be a star forest, where with and . If is an -free bipartite graph of order , where , then with equality if and only if .
Corollary 1.4 Let be a star forest, where with and . If is an -free graph of order , where , then with equality if and only if .
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 for .
Lemma 2.1 Let be a star forest, where with and . If is an -free graph of order with , then
Proof Let . Since is -free, by the definition of . Hence
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 be a star forest, where with and . If is an -free connected graph of order , where , then with equality if and only if , where is a -regular graph of order .
Proof Let 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 be a star forest, where with and . If is an -free connected bipartite graph of order , where , then with equality if and only if .
Proof Let be a connected graph with the maximum spectral radius among all -free connected graphs of order . 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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