[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectivel...[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectively.The largest eigenvalue of A_α(G) is called the A_α-spectral radius of G,denoted by ρ_α(G).In this paper,we give an upper bound on ρ_α(G) of a Hamiltonian graph G with m edges for α∈[1/2,1),and completely characterize the corresponding extremal graph in the case when m is odd.In order to complete the proof of the main result,we give a sharp upper bound on the ρ_α(G) of a connected graph G in terms of its degree sequence.展开更多
Given a graph G,the adjacency matrix and degree diagonal matrix of G are denoted by A(G)and D(G),respectively.In 2017,Nikiforov~([24])proposed the A_(α)-matrix:A_α(G)=αD(G)+(1-α)A(G),whereα∈[0,1].The largest eig...Given a graph G,the adjacency matrix and degree diagonal matrix of G are denoted by A(G)and D(G),respectively.In 2017,Nikiforov~([24])proposed the A_(α)-matrix:A_α(G)=αD(G)+(1-α)A(G),whereα∈[0,1].The largest eigenvalue of this novel matrix is called the A_(α)-index of G.In this paper,we characterize the graphs with minimum A_(α)-index among n-vertex graphs with independence number i forα∈[0,1),where i=1,[n/2],[n/2],[n/2]+1,n-3,n-2,n-1,whereas for i=2 we consider the same problem forα∈[0,3/4].Furthermore,we determine the unique graph(resp.tree)on n vertices with given independence number having the maximum A_(α)-index withα∈[0,1),whereas for the n-vertex bipartite graphs with given independence number,we characterize the unique graph having the maximum A_α-index withα∈[1/2,1).展开更多
文摘[App1.Anal.Discrete Math.,2017,11(1):81-107] defined the A_α-matrix of a graph G as A_α(G)=αD(G)+(1-α)A(G),where α∈[0,1],D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix of G,respectively.The largest eigenvalue of A_α(G) is called the A_α-spectral radius of G,denoted by ρ_α(G).In this paper,we give an upper bound on ρ_α(G) of a Hamiltonian graph G with m edges for α∈[1/2,1),and completely characterize the corresponding extremal graph in the case when m is odd.In order to complete the proof of the main result,we give a sharp upper bound on the ρ_α(G) of a connected graph G in terms of its degree sequence.
基金the Undergraduate Innovation and Entrepreneurship Grant from Central China Normal University(Grant No.20210409037)by Industry-University-Research Innovation Funding of Chinese University(Grant No.2019ITA03033)by the National Natural Science Foundation of China(Grant Nos.12171190,11671164)。
文摘Given a graph G,the adjacency matrix and degree diagonal matrix of G are denoted by A(G)and D(G),respectively.In 2017,Nikiforov~([24])proposed the A_(α)-matrix:A_α(G)=αD(G)+(1-α)A(G),whereα∈[0,1].The largest eigenvalue of this novel matrix is called the A_(α)-index of G.In this paper,we characterize the graphs with minimum A_(α)-index among n-vertex graphs with independence number i forα∈[0,1),where i=1,[n/2],[n/2],[n/2]+1,n-3,n-2,n-1,whereas for i=2 we consider the same problem forα∈[0,3/4].Furthermore,we determine the unique graph(resp.tree)on n vertices with given independence number having the maximum A_(α)-index withα∈[0,1),whereas for the n-vertex bipartite graphs with given independence number,we characterize the unique graph having the maximum A_α-index withα∈[1/2,1).
