In this paper, an equivalent condition of a graph G with t （2≤ t ≤n） distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular （necessarily be strongly regular）, an equi...In this paper, an equivalent condition of a graph G with t （2≤ t ≤n） distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular （necessarily be strongly regular）, an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.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
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
A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulan...A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.展开更多
Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues...Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_(p1,p2,···,pr)=K_(a1·p1,a2·p2,···,as···ps) to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_(a1·p1,a2·p2,···,as·ps) with s > 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with s = 5, 6. The problem of the existence of such distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with arbitrarily large number s remains open.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
We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove tha...We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.展开更多
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
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
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.
基金Supported by the Anhui Provincial Natural Science Foundation(050460102)National Natural Science Foundation of China(10601001,10571163)+3 种基金NSF of Department of Education of Anhui Province(2004kj027,2005kj005zd)Foundation of Anhui Institute of Architecture and Industry(200510307)Foundation of Mathematics Innovation Team of Anhui UniversityFoundation of Talents Group Construction of Anhui University
文摘In this paper, an equivalent condition of a graph G with t （2≤ t ≤n） distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular （necessarily be strongly regular）, an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.
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 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 the National Natural Science Foundation of China(2 0 0 0 CG0 1 0 3) the Fund of"The Developing Program for Outstanding Person"in NPUS & T Innovation Foundation for Young Teachers of Northwestern Polytechnical University.
文摘In this paper, the spectrum and characteristic polynomial for a special kind of symmetric block circulant matrices are given.
基金Foundation item： Supported by Hunan Provincial Natural Science Foundation（13JJ3118）
文摘A graph is called an integral graph if it has an integral spectrum i.e.,all eigenvalues are integers.A graph is called circulant graph if it is Cayley graph on the circulant group,i.e.,its adjacency matrix is circulant.The rank of a graph is defined to be the rank of its adjacency matrix.This importance of the rank,due to applications in physics,chemistry and combinatorics.In this paper,using Ramanujan sums,we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.
基金Supported by the National Natural Science Foundation of China（11171273） Supported by the Graduate Starting Seed Fund of Northwestern Polytechnical University（Z2014173）
文摘Let D(G) =(d_(ij))_(n×n) denote the distance matrix of a connected graph G with order n, where d_(ij) is equal to the distance between vertices viand vjin G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a sufficient and necessary condition for complete r-partite graphs K_(p1,p2,···,pr)=K_(a1·p1,a2·p2,···,as···ps) to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs K_(a1·p1,a2·p2,···,as·ps) with s > 4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with s = 5, 6. The problem of the existence of such distance integral graphs K_(a1·p1,a2·p2,···,as·ps) with arbitrarily large number s remains open.
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the financial support from the Slovenian Research Agency(research core funding No.P1-0222).
文摘We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.
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.
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.
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.
