A graph G is called integral if all the eigenvalues of the adjacency matrix A（G） of G are integers. In this paper, the graphs G4（a, b） and Gs（a, b） with 2a ＋ 6b vertices are defined. We give their characteristi...A graph G is called integral if all the eigenvalues of the adjacency matrix A（G） of G are integers. In this paper, the graphs G4（a, b） and Gs（a, b） with 2a ＋ 6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the （n ＋ 2）-regular graphs G4（n, n＋ 2） and G5（n, n ＋ 2） are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.展开更多
Graphs for the design of networks or electronic circuits are usually weighted and the spectrum of weighted graphs are often analyzed to solve problems. This paper discusses the spectrum and the spectral radii of tree...Graphs for the design of networks or electronic circuits are usually weighted and the spectrum of weighted graphs are often analyzed to solve problems. This paper discusses the spectrum and the spectral radii of trees with edge weights. We derive expressions for the spectrum and the spectral radius of a weighted star, together with the boundary limits of the spectral radii for weighted paths and weighted trees. The analysis uses the theory of nonnegative matrices and applies the 'moving edge' technique. Some simple examples of weighted paths and trees are presented to explain the results. Then, we propose some open problems in this area.展开更多
基金Supported by the National Natural Science Foundation of China (10871158, 70871098)the Natural Science Basic Research Plan in Shaanxi Province of China (SJ08A01, 2007A09) and SRF for ROCS, SEM
文摘A graph G is called integral if all the eigenvalues of the adjacency matrix A（G） of G are integers. In this paper, the graphs G4（a, b） and Gs（a, b） with 2a ＋ 6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the （n ＋ 2）-regular graphs G4（n, n＋ 2） and G5（n, n ＋ 2） are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.
基金supported by NNSF of China(10861009)the State Ethnic Affairs Commission Foundation of China(10QH01)+1 种基金Chunhui of Ministry of Education Project(Z2011014)Qinghai Provincial Natural Science Foundation(2011-Z-911)~~
基金Supported by the National Basic Research PrioritiesProgram(No.G19990 32 90 3
文摘Graphs for the design of networks or electronic circuits are usually weighted and the spectrum of weighted graphs are often analyzed to solve problems. This paper discusses the spectrum and the spectral radii of trees with edge weights. We derive expressions for the spectrum and the spectral radius of a weighted star, together with the boundary limits of the spectral radii for weighted paths and weighted trees. The analysis uses the theory of nonnegative matrices and applies the 'moving edge' technique. Some simple examples of weighted paths and trees are presented to explain the results. Then, we propose some open problems in this area.
