F. Lazebnik, V.A. Ustimenko, and A.J. Woldar. A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series), 32(1):73--79, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....=2) m n log(n ) In particular, Lemma 2.10 implies that, for every s g 1, s;0) G n ) m(g; n) n O(1) where m(g; n) denotes the maximum number of edges in an n node graphs of girth g. In [ERS66] it is conjectured that m(2k 2; n) 1 1=k ) proving later for k = 1; 2; 3 and 5. In [LUW95], it is shown that m(4k 2; n) 1 1= 3k 1) and that m(4k; n) 1 1= 3(k 1) It follows, Theorem 2.11 For every s 1, s;0) G n ) 1= 3s=4 O(1) and (s;0) G n ) 1 1= 3s=4 O(1) 2.4 A Lower Bound on Trees In this section we prove two lower bounds on trees. ....

F. Lazebnik, V. A. Ustimenko, and A. J. Woldar. A new series of dense graphs of high girth. Bulletin of American Mathematical Society (New Series), 32(1):73{ 79, 1995.

....(SG ) m=n O(1) In particular, Lemma 5 implies that, for every s g 1, s;0) Gn ) m(g; n) n O(1) where m(g; n) denotes the maximum number of edges in an n node graphs of girth g. In [ERS66] it is conjectured that m(2k 2; n) n 1 1=k ) proving later for k = 1; 2; 3 and 5. In [LUW95], it is shown that m(4k 2; n) n 1 1= 3k 1) and that m(4k; n) n 1 1= 3(k 1) Therefore, Theorem 3. For every s 1, s;0) Gn ) n 1= 3s=4 O(1) 3 Additive Approximate Schemes 3.1 A Scheme for k chordal Graphs A graph is k chordal if it does not contain any ....

F. Lazebnik, V. A. Ustimenko, and A. J. Woldar. A new series of dense graphs of high girth. Bulletin of American Mathematical Society (New Series) , 32(1):73-79, 1995.

....algebraic techniques, and yield the so called Ramanujan graphs, which are graphs on m vertices with at least Omega i m (3k 5) 3k 3) j edges which do not contain any cycle of length smaller than 2k 1. i.e. N(m; 2k; 2) Omega i n (3k 5) 3k 3) j . Recently Lazebnik, Ustimenko and Woldar [12] showed that N(m; 2k; 2) Omega i m (3k Gamma1 fi) 3k Gamma3 fi) j with fi = 0 if k is odd and fi = 1 otherwise. Here, we do not focus on the case r = 2; we consider the case of arbitrary positive integers k and r. As usual, let gcd(k; l) be the greatest common divisor of positive ....

F. Lazebnik, V. A. Ustimenko and A. J. Woldar, A New Series of Dense Graphs of High Girth, Bulletin (New Series) of the American Mathematical Society 32, 1995, 73-79.

.... Phillips and Sarnak [LPS 88] and Margulis [Ma 88] have improved the probabilistic lower bound to ex(m; fC 3 ; C 2k g) Omega Gamma m 3k 5 3k 3 ) hence, N(m; 2k; 2) c Delta n 3k 5 3k 3 = c Delta n 1 2 3k 3 : Recent constructive improvements by Lazebnik, Ustimenko and Woldar [LUW 95] on the lower bound on ex(m; fC 3 ; C 2k g) imply N(m; 2k; 2) Omega i m 1 2 3k Gamma3 j ; where 2 f0; 1g, and = 0 if and only if k is an odd integer. 5 Explicit Constructions Construction 1. First we will give a construction, which yields N(m; 4; 3) 1=8 Gamma o(1) ....

F. Lazebnik, V. A. Ustimenko, and A. J. Woldar, A New Series of Dense Graphs of High Girth, Bulletin (New Series) of the American Mathematical Society 32, 1995, 73-79.

....p 1 ) # 3 4. The fact that the value of c(L p 1 )is exactly 3 4 was established independently by Margulis [34] and Biggs and Boshier [9] The basic idea of [32] is to use quaternion algebras, and this was extended to cubic graphs by Chiu [16] Recently, Lazebnik, Ustimenko and Woldar [29, 30] have constructed families G k such c(G k ) 3 4) log k 1 k for every k # 3 Unfortunately, their results are weakest for k = 3, since the value of c is then (3 4) log 2 3=1.19. We began with the naive lower bound # 0 (g) # 0 (g) The families mentioned above provide upper bounds for some ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A new series of dense graphs of high girth, Bulletin of the AMS 32 (1995), 73--79

.... lower bound ex(n; C 2t ) Omega Gamma n 1 1=2t ) In the remaining cases the best lower bound is given by the so called Ramanujan graphs of Lubotzky, Phillips and Sarnak [35] Margulis [37] and Imrich [31] Their construction was slightly improved by Lazebnik, Ustimenko and Woldar [34], who borrowed ideas from the theory of Lie algebras to prove that cn 1 2 3k Gamma2 ex(n; C 2k ) On the other hand, in the polynomial range a result of Kov ari, T. S os and Tur an [32] implies that ex(n; k; bk 2 =4c Gamma 1) ex(n; K bk=2c;dk=2e ) O(n 2 Gamma2=k ) Finally, in the ....

F. Lazebnik, V. A. Ustimenko and A. J. Woldar, A new series of dense graphs of high girth, Bull. Amer. Math. Soc. 32 (1995), 73-79.

....than Moore graphs are known (see [Won] HS, Ch. 6] and [Big93, Ch. 23] for surveys of known results) For cubic graphs, f(3; g) is known for 3 g 12 ( BMS] For k 4, the f(k; 7) is already unknown. The best upper bounds for f(k; g) come from various infinite series of symmetric graphs (cf. [LPS, Mar88, Mor, Chi, BH, LUW]) It is known that 1=2 lim inf g 1 log k Gamma1 f(k; g) g lim sup g 1 log k Gamma1 f(k; g) g 3=4 (7) For all known infinite families of regular graphs, lim sup g 1 (log k Gamma1 f(k; g) g = 3=4 it is sometimes conjectured that the upper bound in (7) is actually an asymptotic result. ....

F. Lazebnik, V. Ustimenko and A. Wodlar. A new series of dense graphs of high girth. AMS Bull, 32:73--79, 1995.

....that c(L p 1 ) 3=4. The fact that the value of c(L p 1 ) is exactly 3=4 was established independently by Margulis [34] and Biggs and Boshier [9] The basic idea of [32] is to use quaternion algebras, and this was extended to cubic graphs by Chiu [16] Recently, Lazebnik, Ustimenko and Woldar [29, 30] have constructed families G k such c(G k ) 3=4) log k Gamma1 k for every k 3 Unfortunately, their results are weakest for k = 3, since the value of c is then (3=4) log 2 3 = 1:19: We began with the naive lower bound 0 (g) 0 (g) The families mentioned above provide upper bounds for ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A new series of dense graphs of high girth, Bulletin of the AMS 32 (1995), 73--79

....Let the set of vertices of D(n, q) be Pn Ln . Adjacency in D(n, q) is now defined in terms of the first n 1 relations of (6) and no others. Note that these relations involve only the first n coordinates of vectors from P L, so apply unambiguously to vectors from P n L n . In [12], Lazebnik, Ustimenko and Woldar showed that the graphs D(n, q) are disconnected for n 6 and that their connected components CD(n, q) all isomorphic) have order at most 2q # 1 . As CD(n, q) and D(n, q) have the same girth, it follows that the graphs CD(n, q) form a family for which # ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A new series of dense graphs of high girth, Bulletin of the AMS 32 (1) (1995), 73-79.

....Let the set of vertices of D(n; q) be Pn [ Ln . Adjacency in D(n; q) is now defined in terms of the first n Gamma 1 relations of (6) and no others. Note that these relations involve only the first n coordinates of vectors from P [ L, so apply unambiguously to vectors from Pn [ Ln . In [12], Lazebnik, Ustimenko and Woldar showed that the graphs D(n; q) are disconnected for n 6 and that their connected components CD(n; q) all isomorphic) have order at most 2q n Gammab n 2 4 c 1 . As CD(n; q) and D(n; q) have the same girth, it follows that the graphs CD(n; q) form a family for ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A new series of dense graphs of high girth, Bulletin of the AMS 32 (1) (1995), 73-79.

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F. Lazebnik, V.A. Ustimenko, and A.J. Woldar. A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series), 32(1):73--79, 1995.

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F. Lazebnik, V. Ustimenko, and A. Woldar. A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series), 32(1):73--79, 1995.

No context found.

F. Lazebnik, V.A. Ustimenko, and A.J. Woldar. A new series of dense graphs of high girth. Bulletin of the American Mathematical Society (New Series), 32(1):73--79, 1995.

No context found.

F. Lazebnik, V.A. Ustimenko, and A.J. Woldar, A new series of dense graphs of high girth, Bulletin of the AMS 32 (1995), no. 1, 73--79.

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F. Lazebnik, V. A. Ustimenko, and A. J. Woldar. A New Series of Dense Graphs of High Girth. Bulletin of the American Mathematical Society (N.S.), 32(1):73--79, 1995.

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