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
Let G be a graph and A(G) the adjacency matrix of G. The spectrum of G is the eigenvalues together with their multiplicities of A(G). Chang et al. (2011) characterized the structures of all graphs with rank 4. Monsalv...Let G be a graph and A(G) the adjacency matrix of G. The spectrum of G is the eigenvalues together with their multiplicities of A(G). Chang et al. (2011) characterized the structures of all graphs with rank 4. Monsalve and Rada (2021) gave the bound of spectral radius of all graphs with rank 4. Based on these results as above, we further investigate the spectral properties of graphs with rank 4. And we give the expressions of the spectral radius and energy of all graphs with rank 4. In particular, we show that some graphs with rank 4 are determined by their spectra.展开更多
A graph G is said to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.We consider θ-graphs,that is,graphs obtained by subdividing the edges of the multig...A graph G is said to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.We consider θ-graphs,that is,graphs obtained by subdividing the edges of the multigraph consist of three parallel edges.In this paper,some special θ-graphs are determined by their Laplacian spectra.展开更多
Let Hn（p,q） be a tree obtained from two stars K1,p and K1,q by identifying the center of K1,p with one end of a path Pn and the center of K1,q with the other end of Pn.We call Hn（p,p-1） a double quasi-star tree.In...Let Hn（p,q） be a tree obtained from two stars K1,p and K1,q by identifying the center of K1,p with one end of a path Pn and the center of K1,q with the other end of Pn.We call Hn（p,p-1） a double quasi-star tree.In this paper,we show that a double quasi-star tree is determined by its Laplacian spectrum.展开更多
Two nonisomorphic graphs G and H are said to be matching equivalent if and only if G and H have the same matching polynomials. In this paper, some families matching equivalent graphs are constructed. In particular, a ...Two nonisomorphic graphs G and H are said to be matching equivalent if and only if G and H have the same matching polynomials. In this paper, some families matching equivalent graphs are constructed. In particular, a new method to construct cospectral forests is given.展开更多
A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. An H-shape is a tree with exactly two of its vertices having maximal degree 3. In this paper, a formula...A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. An H-shape is a tree with exactly two of its vertices having maximal degree 3. In this paper, a formula of counting the number of closed 6-walks is given on a graph, and some necessary conditions of a graph Γ cospectral to an H-shape are given.展开更多
In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes a...In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes are infinite. The discovery of these classes is a new contribution to the search of such integral graphs.展开更多
Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By th...Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.展开更多
In this paper, we show that some edges-deleted subgraphs of complete graph are determined by their spectrum with respect to the adjacency matrix as well as the Laplacian matrix.
Given graphs Gand G, we define a graph operation on Gand G,namely the SSG-vertex join of Gand G, denoted by G★ G. Let S(G) be the subdivision graph of G. The SSG-vertex join G★Gis the graph obtained from S(G) and S(...Given graphs Gand G, we define a graph operation on Gand G,namely the SSG-vertex join of Gand G, denoted by G★ G. Let S(G) be the subdivision graph of G. The SSG-vertex join G★Gis the graph obtained from S(G) and S(G) by joining each vertex of Gwith each vertex of G. In this paper, when G(i = 1, 2) is a regular graph, we determine the normalized Laplacian spectrum of G★ G. As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of G★G.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.
文摘Let G be a graph and A(G) the adjacency matrix of G. The spectrum of G is the eigenvalues together with their multiplicities of A(G). Chang et al. (2011) characterized the structures of all graphs with rank 4. Monsalve and Rada (2021) gave the bound of spectral radius of all graphs with rank 4. Based on these results as above, we further investigate the spectral properties of graphs with rank 4. And we give the expressions of the spectral radius and energy of all graphs with rank 4. In particular, we show that some graphs with rank 4 are determined by their spectra.
基金National Natural Science Foundation of China (No. 11071078,No. 11075057 )Open Research Funding Program of LGISEM and Shanghai Leading Academic Discipline Project,China (No. B407)
文摘A graph G is said to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.We consider θ-graphs,that is,graphs obtained by subdividing the edges of the multigraph consist of three parallel edges.In this paper,some special θ-graphs are determined by their Laplacian spectra.
基金Project supported by the Natural Science Foundation of Gausu Province (Grant Nos.3Z5051-A25-037, 0809RJZA017)the National Natural Science Foundation of China (Grant No.50877034)the Foundation of Lanzhou University of Technology(Grant No.0914ZX136)
文摘Let Hn（p,q） be a tree obtained from two stars K1,p and K1,q by identifying the center of K1,p with one end of a path Pn and the center of K1,q with the other end of Pn.We call Hn（p,p-1） a double quasi-star tree.In this paper,we show that a double quasi-star tree is determined by its Laplacian spectrum.
文摘Two nonisomorphic graphs G and H are said to be matching equivalent if and only if G and H have the same matching polynomials. In this paper, some families matching equivalent graphs are constructed. In particular, a new method to construct cospectral forests is given.
文摘A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. An H-shape is a tree with exactly two of its vertices having maximal degree 3. In this paper, a formula of counting the number of closed 6-walks is given on a graph, and some necessary conditions of a graph Γ cospectral to an H-shape are given.
文摘In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes are infinite. The discovery of these classes is a new contribution to the search of such integral graphs.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671344 and 11531011)
文摘Let p be an odd prime, and D2p = （a,b I aP = b2 = l,bab= a 1） the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D2 by using Gauss＇ celebrated law of quadratic reciprocity.
基金Supported by the National Natural Science Foundation of China (Grant No10861009)the State Ethnic Affairs Commission Foundation of China (Grant No09QH02)
文摘In this paper, we show that some edges-deleted subgraphs of complete graph are determined by their spectrum with respect to the adjacency matrix as well as the Laplacian matrix.
文摘Given graphs Gand G, we define a graph operation on Gand G,namely the SSG-vertex join of Gand G, denoted by G★ G. Let S(G) be the subdivision graph of G. The SSG-vertex join G★Gis the graph obtained from S(G) and S(G) by joining each vertex of Gwith each vertex of G. In this paper, when G(i = 1, 2) is a regular graph, we determine the normalized Laplacian spectrum of G★ G. As applications, we construct many pairs of normalized Laplacian cospectral graphs, the normalized Laplacian energy, and the degree Kirchhoff index of G★G.
