LetΓdenote a bipartite Q-polynomial distance-regular graph with vertex set X,valency k≥3 and diameter D≥3.Let A be the adjacency matrix ofΓand let A*:=A*(x)be the dual adjacency matrix ofΓwith respect to a fixed ...LetΓdenote a bipartite Q-polynomial distance-regular graph with vertex set X,valency k≥3 and diameter D≥3.Let A be the adjacency matrix ofΓand let A*:=A*(x)be the dual adjacency matrix ofΓwith respect to a fixed vertex x∈X.Let T:=T(x)denote the Terwilliger algebra ofΓgenerated by A and A*.In this paper,we first describe the relations between A and A*.Then we determine the dimensions of both T and the center of T,and moreover we give a basis of T.展开更多
It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q...It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q-polynomial structures are either dual bipartite or almost dual bipartite. By the work of Dickie(1995) this implies that any distance-regular graph with diameter d at least four and valency at least three admitting two Q-polynomial structures is, provided it is not a Hadamard graph, either the cube H(d, 2)with d even, the half cube 1/2H(2d + 1, 2), the folded cube?H(2d + 1, 2), or the dual polar graph on [2A2d-1(q)]with q 2 a prime power.展开更多
Recently,Makhnev and Nirova found intersection arrays of distance-regular graphs withλ=2 and at most 4096 vertices.In the case of primitive graphs of diameter 3 withμ=1 there corresponding arrays are{18,15,9;1,1,10}...Recently,Makhnev and Nirova found intersection arrays of distance-regular graphs withλ=2 and at most 4096 vertices.In the case of primitive graphs of diameter 3 withμ=1 there corresponding arrays are{18,15,9;1,1,10},{33,30,8;1,1,30}or{39,36,4;1,1,36}.In this work,possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array{18,15,9;1,1,10}are studied.In particular,graph with intersection array{18,15,9;1,1,10}is not vertex symmetric.展开更多
Earlier it was proved that some distance-regular graphs of diameter 3 with c_(2)=2 do not exist.Distance-regular graphΓwith intersection array{17,16,10;1,2,8}has strongly regular graphΓ_(3)(pseudo-geometric graph fo...Earlier it was proved that some distance-regular graphs of diameter 3 with c_(2)=2 do not exist.Distance-regular graphΓwith intersection array{17,16,10;1,2,8}has strongly regular graphΓ_(3)(pseudo-geometric graph for the net pG_(9)(17,9)).By symmetrizing the arrays of triple intersection numbers,it is proved that the distanceregular graphs with intersection arrays{17,16,10;1,2,8}and{22,21,4;1,2,14}do not exist.展开更多
Let D be an integer at least 3 and let H(D, 2) denote the hypercube. It is known that H(D, 2) is a Q-polynomial distance-regular graph with diameter D, and its eigenvalue sequence and its dual eigenvalue sequence are ...Let D be an integer at least 3 and let H(D, 2) denote the hypercube. It is known that H(D, 2) is a Q-polynomial distance-regular graph with diameter D, and its eigenvalue sequence and its dual eigenvalue sequence are all {D-2i}D i=0. Suppose that denotes the tetrahedron algebra. In this paper, the authors display an action of ■ on the standard module V of H(D, 2). To describe this action, the authors define six matrices in Mat X(C), called A, A*, B, B*, K, K*.Moreover, for each matrix above, the authors compute the transpose and then compute the transpose of each generator of ■ on V.展开更多
Let Г be a distance-regular graph of diameter 3 with strong regular graph Г_(3).The determination of the parameters Г_(3) over the intersection array of the graph Г is a direct problem.Finding an intersection arra...Let Г be a distance-regular graph of diameter 3 with strong regular graph Г_(3).The determination of the parameters Г_(3) over the intersection array of the graph Г is a direct problem.Finding an intersection array of the graph Г with respect to the parameters Г_(3) is an inverse problem.Previously,inverse problemswere solved for Г_(3) by Makhnev and Nirova.In this paper,we study the intersection arrays of distance-regular graph Г of diameter 3,for which the graph Г_(3) is a pseudo-geometric graph of the net PGm(n,m).New infinite series of admissible intersection arrays for these graphs are found.We also investigate the automorphisms of distance-regular graph with the intersection array{20,16,5;1,1,16}.展开更多
Let F denote the folded （2D ＋ 1）-cube with vertex set X and diameter D ≥ 3. Fix x∈ X. We first define a partial order ≤ on X as follows. For y,z ∈ X let y ≤ z whenever （x,y）＋ （y,z） =- （x, z）. Let R ...Let F denote the folded （2D ＋ 1）-cube with vertex set X and diameter D ≥ 3. Fix x∈ X. We first define a partial order ≤ on X as follows. For y,z ∈ X let y ≤ z whenever （x,y）＋ （y,z） =- （x, z）. Let R （resp. L） denote the raising matrix （resp. lowering matrix） of P. Next we show that there exists a certain linear dependency among RL2, LRL, L2R and L for each given Q-polynomial structure of F. Finally, we determine whether the above linear dependency structure gives this poser a uniform structure or strongly uniform structure.展开更多
基金supported by the National Natural Science Foundation of China(No.11971146 and No.12101175)the Natural Science Foundation of Hebei Province(No.A2020403024 and No.A2023205009).
文摘LetΓdenote a bipartite Q-polynomial distance-regular graph with vertex set X,valency k≥3 and diameter D≥3.Let A be the adjacency matrix ofΓand let A*:=A*(x)be the dual adjacency matrix ofΓwith respect to a fixed vertex x∈X.Let T:=T(x)denote the Terwilliger algebra ofΓgenerated by A and A*.In this paper,we first describe the relations between A and A*.Then we determine the dimensions of both T and the center of T,and moreover we give a basis of T.
基金supported by Natural Science Foundation of Hebei Province(Grant No.A2012205079)Science Foundation of Hebei Normal University(Grant No.L2011B02)the 100 Talents Program of the Chinese Academy of Sciences for support
文摘It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q-polynomial structures are either dual bipartite or almost dual bipartite. By the work of Dickie(1995) this implies that any distance-regular graph with diameter d at least four and valency at least three admitting two Q-polynomial structures is, provided it is not a Hadamard graph, either the cube H(d, 2)with d even, the half cube 1/2H(2d + 1, 2), the folded cube?H(2d + 1, 2), or the dual polar graph on [2A2d-1(q)]with q 2 a prime power.
基金The work was performed under support of RSF,project 14-11-00061(Theorem 1.1)agreement between ministry of education and science of Russian Federation and the Ural federal university on 27.08.2013,No.02.A03.21.0006(Corollary 1.2).
文摘Recently,Makhnev and Nirova found intersection arrays of distance-regular graphs withλ=2 and at most 4096 vertices.In the case of primitive graphs of diameter 3 withμ=1 there corresponding arrays are{18,15,9;1,1,10},{33,30,8;1,1,30}or{39,36,4;1,1,36}.In this work,possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array{18,15,9;1,1,10}are studied.In particular,graph with intersection array{18,15,9;1,1,10}is not vertex symmetric.
基金supported by the RFBR and the NFSC(Project No.20-51-53013)supported by the NNSF of China(No.12171126).
文摘Earlier it was proved that some distance-regular graphs of diameter 3 with c_(2)=2 do not exist.Distance-regular graphΓwith intersection array{17,16,10;1,2,8}has strongly regular graphΓ_(3)(pseudo-geometric graph for the net pG_(9)(17,9)).By symmetrizing the arrays of triple intersection numbers,it is proved that the distanceregular graphs with intersection arrays{17,16,10;1,2,8}and{22,21,4;1,2,14}do not exist.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10171006) the Youth Scientific Foundation of Beijing Normal University.
文摘In this paper, it is proved that the girth of a 4-homogeneous bipartite graph with valency greaterthan 2 is at most 12.
基金supported by the National Natural Science Foundation of China(Nos.11471097,11271257)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20121303110005)+1 种基金the Natural Science Foundation of Hebei Province(No.A2013205021)the Key Fund Project of Hebei Normal University(No.L2012Z01)
文摘Let D be an integer at least 3 and let H(D, 2) denote the hypercube. It is known that H(D, 2) is a Q-polynomial distance-regular graph with diameter D, and its eigenvalue sequence and its dual eigenvalue sequence are all {D-2i}D i=0. Suppose that denotes the tetrahedron algebra. In this paper, the authors display an action of ■ on the standard module V of H(D, 2). To describe this action, the authors define six matrices in Mat X(C), called A, A*, B, B*, K, K*.Moreover, for each matrix above, the authors compute the transpose and then compute the transpose of each generator of ■ on V.
基金supported by RSF,project 14-11-00061-Пsupported by the NNSF of China(11771409).
文摘Let Г be a distance-regular graph of diameter 3 with strong regular graph Г_(3).The determination of the parameters Г_(3) over the intersection array of the graph Г is a direct problem.Finding an intersection array of the graph Г with respect to the parameters Г_(3) is an inverse problem.Previously,inverse problemswere solved for Г_(3) by Makhnev and Nirova.In this paper,we study the intersection arrays of distance-regular graph Г of diameter 3,for which the graph Г_(3) is a pseudo-geometric graph of the net PGm(n,m).New infinite series of admissible intersection arrays for these graphs are found.We also investigate the automorphisms of distance-regular graph with the intersection array{20,16,5;1,1,16}.
基金Supported by the Natural Science Foundation of China(No.11471097)the Innovative Fund Project of Hebei Province(sj.2017084)
文摘Let F denote the folded （2D ＋ 1）-cube with vertex set X and diameter D ≥ 3. Fix x∈ X. We first define a partial order ≤ on X as follows. For y,z ∈ X let y ≤ z whenever （x,y）＋ （y,z） =- （x, z）. Let R （resp. L） denote the raising matrix （resp. lowering matrix） of P. Next we show that there exists a certain linear dependency among RL2, LRL, L2R and L for each given Q-polynomial structure of F. Finally, we determine whether the above linear dependency structure gives this poser a uniform structure or strongly uniform structure.
