In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue ...In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.展开更多
Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,re...Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,respectively.In this paper,we investigate how the least eigenvalue of A_(α)(G)changes when the graph G is perturbed by deleting a vertex,subdividing edges or moving edges,respectively.展开更多
Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,whe...Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.展开更多
Let U^(g)_(n)be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),…,T_(g))∈U^(g)_(n)be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in an anticlockwise direction)by identifying v_(i)with...Let U^(g)_(n)be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),…,T_(g))∈U^(g)_(n)be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in an anticlockwise direction)by identifying v_(i)with the root of a rooted tree T_(i)of order n_(i)for each i=1,2,…,g,where n_(i)≥1 and∑^(g)_(i=1)n_(i)=n.In this note,the graph with the minimal least eigenvalue(and the graph with maximal spread)in C(T_(1),T_(2),…,T_(g))is determined.展开更多
Let G = （V（G）, E（G）） be a simple connected graph of order n. For any vertices u, v, w ￠V（G） with uv ∈ E（G） and uw ￠ E（G）, an edge-rotating of G means rotating the edge uv （around u） to the non-edge po...Let G = （V（G）, E（G）） be a simple connected graph of order n. For any vertices u, v, w ￠V（G） with uv ∈ E（G） and uw ￠ E（G）, an edge-rotating of G means rotating the edge uv （around u） to the non-edge position uw. In this work, we consider how the least eigenvalue of a graph perturbs when the graph is performed by rotating an edge from the shorter hanging path to the longer one.展开更多
The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix.We characterize the connected graphs of order n and size n+k(5≤k≤8 and n≥k+5)with the minimal least eigenvalue.
A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Lapl...A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.展开更多
A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k.The well-known Brualdi-Solheid problem asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the...A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k.The well-known Brualdi-Solheid problem asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of extremal graphs.In this paper,we first characterize k-degenerate graphs with the minimum least eigenvalue.As an application,we give an answer to the Brualdi-Solheid problem for k-degenerate bipartite graphs.In addition,we give a new proof to determine the maximum signless Laplacian spectral radius and characterize the corresponding extremal graphs among k-degenerate graphs.We also raise a problem related to the second largest eigenvalues of k-degenerate graphs and solve this problem for 1-degenerate graphs.展开更多
Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are intereste...Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue.With the finite element discretization,we propose a monotonically decreasing algorithm to solve the minimization problem.Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities.As the computations are sensitive to the choice of the discretization mesh sizes,we adopt the refined mesh strategy,whose mesh grids are 25-times of the amount used in[S.Osher and F.Santosa,J.Comput.Phys.,171(2001),pp.272-288].We also show the significant reduction in computational cost with the fast convergence of this algorithm.展开更多
The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ...The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ｜V （G）｜ ＋ 1. Let B（n, g） be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B（n, g） is determined.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11071002)Program for New Century Excellent Talents in University,Key Project of Chinese Ministry of Education(Grant No.210091)+8 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20103401110002)Science and Technological Fund of Anhui Province for Outstanding Youth(Grant No.10040606Y33)the Natural Science Foundation of Department of Education of Anhui Province(Grant Nos.KJ2011A195KJ2010B136)Project of Anhui Province for Excellent Young Talents in Universities(Grant No.2009SQRZ017ZD)Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(Grant No.KJJQ1001)Project for Academic Innovation Team of Anhui University(Grant No.KJTD001B)Fund for Youth Scientific Research of Anhui University(Grant No.KJQN1003)Innovation Fund for Graduates of Anhui University
文摘In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1207141112171222)the Basic Research Program(Natural Science)of Yancheng(Grant No.YCBK2024043)。
文摘Let G be a simple undirected graph.For any real numberα∈[0,1],Nikiforov defined the A_(α)-matrix of G as A_(α)(G)=αD(G)+(1-α)A(G),where A(G)and D(G)are the adjacency matrix and the degree diagonal matrix of G,respectively.In this paper,we investigate how the least eigenvalue of A_(α)(G)changes when the graph G is perturbed by deleting a vertex,subdividing edges or moving edges,respectively.
基金Supported by the Supported by the National Natural Science Foundation of China (Grant No. 11071002)Key Project of Chinese Ministry of Education (Grant No. 210091)+4 种基金Anhui Provincial Natural Science Foundation(Grant No. 10040606Y33)Anhui University Innovation Team Project (Grant No. KJTD001B)Project of Anhui Province for Young Teachers Research Support in Universities (Grant No. 2008JQl021)Project of Anhui Province for Excellent Young Talents in Universities (Grant No. 2009SQRZ017ZD)the Natural Science Foundation of Department of Education of Anhui Province (Grant No. KJ2010B136)
文摘Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.
文摘Let U^(g)_(n)be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),…,T_(g))∈U^(g)_(n)be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in an anticlockwise direction)by identifying v_(i)with the root of a rooted tree T_(i)of order n_(i)for each i=1,2,…,g,where n_(i)≥1 and∑^(g)_(i=1)n_(i)=n.In this note,the graph with the minimal least eigenvalue(and the graph with maximal spread)in C(T_(1),T_(2),…,T_(g))is determined.
基金Supported by the National Natural Science Foundation of China(No.11201432,No.11301440)the Natural Science Foundation of Education Ministry of Henan Province(No.15A110003,No.15IRTSTHN006,No.13B110939)
文摘Let G = （V（G）, E（G）） be a simple connected graph of order n. For any vertices u, v, w ￠V（G） with uv ∈ E（G） and uw ￠ E（G）, an edge-rotating of G means rotating the edge uv （around u） to the non-edge position uw. In this work, we consider how the least eigenvalue of a graph perturbs when the graph is performed by rotating an edge from the shorter hanging path to the longer one.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11371372).
文摘The least eigenvalue of a connected graph is the least eigenvalue of its adjacency matrix.We characterize the connected graphs of order n and size n+k(5≤k≤8 and n≥k+5)with the minimal least eigenvalue.
基金supported by the NSF of China(No.19971056)SRP(No.03B019) from the Education Committee of Hunan Province
文摘A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.
基金supported by NSFC(Nos.11771141 and 12011530064)the China Scholarship Council(CSC)(No.202106740049)+2 种基金supported by the National Research Foundation of Korea(NRF)grant funded by the Korean government(MSIT)(No.NRF-2020R1A2C1A01101838)supported by NSFC(No.12161141006)CPSF(No.2021M691671).
文摘A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k.The well-known Brualdi-Solheid problem asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of extremal graphs.In this paper,we first characterize k-degenerate graphs with the minimum least eigenvalue.As an application,we give an answer to the Brualdi-Solheid problem for k-degenerate bipartite graphs.In addition,we give a new proof to determine the maximum signless Laplacian spectral radius and characterize the corresponding extremal graphs among k-degenerate graphs.We also raise a problem related to the second largest eigenvalues of k-degenerate graphs and solve this problem for 1-degenerate graphs.
基金supported by the Chinese National Science Foundation(No.10871179)the National Basic Research Programme of China(No.2008CB717806)Specialized Research Fund for the Doctoral Program of Higher Education of China(SRFDP No.20070335201).
文摘Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density.In this paper,we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue.With the finite element discretization,we propose a monotonically decreasing algorithm to solve the minimization problem.Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities.As the computations are sensitive to the choice of the discretization mesh sizes,we adopt the refined mesh strategy,whose mesh grids are 25-times of the amount used in[S.Osher and F.Santosa,J.Comput.Phys.,171(2001),pp.272-288].We also show the significant reduction in computational cost with the fast convergence of this algorithm.
基金Supported by the National Natural Science Foundation of China(No.11101057)China Postdoctoral Science Foundation(No.20110491443)+1 种基金the NSF of Education Ministry of Anhui province(No.KJ2012Z283)Scientific Research Foundation of Chuzhou University(No.2011kj004B)
文摘The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and ｜E（G）｜ = ｜V （G）｜ ＋ 1. Let B（n, g） be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B（n, g） is determined.
