497 research outputs found

    Distance-regular Cayley graphs with small valency

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    We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most 44, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 55, and the Cayley graphs among all distance-regular graphs with girth 33 and valency 66 or 77. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

    On the Expansion of Group-Based Lifts

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    A kk-lift of an nn-vertex base graph GG is a graph HH on nΓ—kn\times k vertices, where each vertex vv of GG is replaced by kk vertices v1,β‹―,vkv_1,\cdots{},v_k and each edge (u,v)(u,v) in GG is replaced by a matching representing a bijection Ο€uv\pi_{uv} so that the edges of HH are of the form (ui,vΟ€uv(i))(u_i,v_{\pi_{uv}(i)}). Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: (1) There is a constant c1c_1 such that for every kβ‰₯2c1ndk\geq 2^{c_1nd}, there does not exist an abelian kk-lift HH of any nn-vertex dd-regular base graph with HH being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most O(d)O(\sqrt{d}) in magnitude). This can be viewed as an analogue of the well-known no-expansion result for abelian Cayley graphs. (2) A uniform random lift in a cyclic group of order kk of any nn-vertex dd-regular base graph GG, with the nontrivial eigenvalues of the adjacency matrix of GG bounded by Ξ»\lambda in magnitude, has the new nontrivial eigenvalues also bounded by Ξ»+O(d)\lambda+O(\sqrt{d}) in magnitude with probability 1βˆ’keβˆ’Ξ©(n/d2)1-ke^{-\Omega(n/d^2)}. In particular, there is a constant c2c_2 such that for every k≀2c2n/d2k\leq 2^{c_2n/d^2}, there exists a lift HH of every Ramanujan graph in a cyclic group of order kk with HH being almost Ramanujan. We use this to design a quasi-polynomial time algorithm to construct almost Ramanujan expanders deterministically. The existence of expanding lifts in cyclic groups of order k=2O(n/d2)k=2^{O(n/d^2)} can be viewed as a lower bound on the order k0k_0 of the largest abelian group that produces expanding lifts. Our results show that the lower bound matches the upper bound for k0k_0 (upto d3d^3 in the exponent)
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