We introduce the notion of a quadratic graph,which is a graph whose eigenvalues are integral or quadratic algebraic integral,and we determine nine infinite families of quadratic starlike trees,which are just all the q...We introduce the notion of a quadratic graph,which is a graph whose eigenvalues are integral or quadratic algebraic integral,and we determine nine infinite families of quadratic starlike trees,which are just all the quadratic starlike trees including integral starlike trees.Thus,the quadratic starlike trees are completely characterized.The expressions for characteristic polynomials of quadratic starlike trees are also given.展开更多
We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having...We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.展开更多
Let n,k and l be integers with 1≤k<l≤n-1.The set-inclusion graph G(n,k,l)is the graph whose vertex set consists of all κ-and l-subsets of[n]={1,2,...,n},where two distinct vertices are adjacent if one of them is...Let n,k and l be integers with 1≤k<l≤n-1.The set-inclusion graph G(n,k,l)is the graph whose vertex set consists of all κ-and l-subsets of[n]={1,2,...,n},where two distinct vertices are adjacent if one of them is contained in the other.In this paper,we determine the spectrum and automorphism group of G(n,k,l).展开更多
Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = （1 2…n）, cn^-1, （1 2）} and Fn=Cay（Sn, S） be the Cayley graph on Sn with respect to S. In this paper, we show that Fn （n 〉 13） is a normal ...Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = （1 2…n）, cn^-1, （1 2）} and Fn=Cay（Sn, S） be the Cayley graph on Sn with respect to S. In this paper, we show that Fn （n 〉 13） is a normal Cayley graph, and that the full automorphism group of Fn is equal to Aut（Гn） = R（Sn） （Inn（C））≌- Sn×Z2, where R（Sn） is the right regular representation of Sn,φ = （1 2）（3 n）（4 n-1）（5 n-2）... （∈ Sn）, and Inn（C） is the inner isomorphism of Sn induced by φ .展开更多
基金Supported by National Natural Science Foundation of China(Nos.11971274,12061074,11671344).
文摘We introduce the notion of a quadratic graph,which is a graph whose eigenvalues are integral or quadratic algebraic integral,and we determine nine infinite families of quadratic starlike trees,which are just all the quadratic starlike trees including integral starlike trees.Thus,the quadratic starlike trees are completely characterized.The expressions for characteristic polynomials of quadratic starlike trees are also given.
基金This work is supported by the National Natural Science Foundation of China (grants No. 11671344, 11531011 and 11701492).
文摘We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
基金The first author was supported by National Natural Science Foundation of China(No.11901540)China Postdoctoral Science Foundation(No.2019M652556)+2 种基金Postdoctoral Research Sponsorship in Henan Province(No.1902011)The second author was supported by National Natural Science Foundation of China(No.11671344)The third author was supported by National Natural Science Foundation of China(No.11971274).
文摘Let n,k and l be integers with 1≤k<l≤n-1.The set-inclusion graph G(n,k,l)is the graph whose vertex set consists of all κ-and l-subsets of[n]={1,2,...,n},where two distinct vertices are adjacent if one of them is contained in the other.In this paper,we determine the spectrum and automorphism group of G(n,k,l).
基金This work is supported by the National Natural Science Foundation of China （Grant Nos. 11671344 and 11531011）.
文摘Let Sn denote the symmetric group of degree n with n 〉 3, S = {cn = （1 2…n）, cn^-1, （1 2）} and Fn=Cay（Sn, S） be the Cayley graph on Sn with respect to S. In this paper, we show that Fn （n 〉 13） is a normal Cayley graph, and that the full automorphism group of Fn is equal to Aut（Гn） = R（Sn） （Inn（C））≌- Sn×Z2, where R（Sn） is the right regular representation of Sn,φ = （1 2）（3 n）（4 n-1）（5 n-2）... （∈ Sn）, and Inn（C） is the inner isomorphism of Sn induced by φ .
