F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k; q), Discrete Math., 157 (1996), 271--283.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....; 2 2 Isow(W ) be weak isomorphisms leaving any cell of W fixed such that ( 1 ) V i = 2 ) V i for all i 2 [s] Then it can be proved by the same technique that 1 Gamma1 2 is induced by a strong isomorphism of W iff t( 1 ) t( 2 ) mod 2) 6.4. Proof of Theorem 1.2. It follows from [10] and [13] that for all sufficiently large l the graph CD(l; 3) defined in [10] is a connected, edge transitive, cubic Ramanujan graph with s l = 2 Delta 3 l Gammab l 2 4 c 1 vertices. One can easily veryfy that there exists 0 0 such that any cubic Ramanujan graph with s vertices has ....

....1 ) V i = 2 ) V i for all i 2 [s] Then it can be proved by the same technique that 1 Gamma1 2 is induced by a strong isomorphism of W iff t( 1 ) t( 2 ) mod 2) 6.4. Proof of Theorem 1.2. It follows from [10] and [13] that for all sufficiently large l the graph CD(l; 3) defined in [10] is a connected, edge transitive, cubic Ramanujan graph with s l = 2 Delta 3 l Gammab l 2 4 c 1 vertices. One can easily veryfy that there exists 0 0 such that any cubic Ramanujan graph with s vertices has no separator of cardinality k for all k 0 s. By Lemma 6.2 there exists a ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k; q), Discrete Math., 157 (1996), 271--283.

....and # 1 ,# 2 # Isow(W ) be weak isomorphisms leaving any cell of W fixed such that (# 1 ) V i = # 2 ) V i for all i # [s] Then it can be proved by the same technique that # 1 # 1 2 is induced by a strong isomorphism of W i# t(# 1 ) t(# 2 ) mod2) 5.4. Proof of Theorem 1.3. It follows from [12] and [14] that for all su#ciently large l the graph CD(l, 3) defined in [12] is a connected, edge transitive, cubic Ramanujan graph with s l =23 l # l 2 4 # 1 vertices. One can easily veryfy that there exists # # 0 such that any cubic Ramanujan graph with s vertices has no separator of ....

....such that (# 1 ) V i = # 2 ) V i for all i # [s] Then it can be proved by the same technique that # 1 # 1 2 is induced by a strong isomorphism of W i# t(# 1 ) t(# 2 ) mod2) 5.4. Proof of Theorem 1.3. It follows from [12] and [14] that for all su#ciently large l the graph CD(l, 3) defined in [12] is a connected, edge transitive, cubic Ramanujan graph with s l =23 l # l 2 4 # 1 vertices. One can easily veryfy that there exists # # 0 such that any cubic Ramanujan graph with s vertices has no separator of cardinality k for all k # # # s. By Lemma 5.2 there exists a cellular ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k,q), Discrete Math., 157 (1996), 271--283.

....; 2 2 Isow(W ) be weak isomorphisms leaving any cell of W fixed such that ( 1 ) V i = 2 ) V i for all i 2 [s] Then it can be proved by the same technique that 1 Gamma1 2 is induced by a strong isomorphism of W iff t( 1 ) t( 2 ) mod 2) 5.4. Proof of Theorem 1.3. It follows from [12] and [14] that for all sufficiently large l the graph CD(l; 3) defined in [12] is a connected, edge transitive, cubic Ramanujan graph with s l = 2 Delta 3 l Gammab l 2 4 c 1 vertices. One can easily veryfy that there exists 0 0 such that any cubic Ramanujan graph with s vertices has no ....

....1 ) V i = 2 ) V i for all i 2 [s] Then it can be proved by the same technique that 1 Gamma1 2 is induced by a strong isomorphism of W iff t( 1 ) t( 2 ) mod 2) 5.4. Proof of Theorem 1.3. It follows from [12] and [14] that for all sufficiently large l the graph CD(l; 3) defined in [12] is a connected, edge transitive, cubic Ramanujan graph with s l = 2 Delta 3 l Gammab l 2 4 c 1 vertices. One can easily veryfy that there exists 0 0 such that any cubic Ramanujan graph with s vertices has no separator of cardinality k for all k 0 s. By Lemma 5.2 there exists a ....

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k; q), Discrete Math., 157 (1996), 271--283.

....in (1) for all k 5. For k 3 and q 1 mod t, k, 2t) graphs of orders at least as large as the upper bound in (2) were constructed in [9] by Furedi, Seress, and the authors. The fact that the orders of these constructions actually meet the upper bound in (2) for q odd follows from [13]. The constructions we introduce in this paper are independent of the relative magnitudes of k and g. Constructions. For all k 6, g even, we explicitly construct a (k, g) graph of order g ; 3) 5, g odd, we explicitly construct a (k, g) graph of order (g 1) 4) In either ....

....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 # 1) With few exceptions, these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. Later, in [13], the authors proved that for q odd, the order of CD(n, q) is exactly 2q # 1 . Combining this with a result from [9] on the existence of a girth cycle of length n 5 in D(n, q) for all odd n and infinitely many q, we get that the corresponding subfamily of graphs CD(n, q) satisfies # = ....

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F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k, q), Discrete Mathematics 157 (1996) 271--283.

....in (1) for all k 5 and g 5. For k t 3 and q j 1 mod t, k; 2t) graphs of orders at least as large as the upper bound in (2) were constructed in [9] by Furedi, Seress, and the authors. The fact that the orders of these constructions actually meet the upper bound in (2) for q odd follows from [13]. The constructions we introduce in this paper are independent of the relative magnitudes of k and g. Constructions. For all k 3 and g 6, g even, we explicitly construct a (k; g) graph of order g i 1 (k Gamma 2)kq g Gamma5 Gammab g Gamma3 4 c j ; 3) for all k 3 and g 5, g odd, ....

..... 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 fl 4 3 log q (q Gamma 1) With few exceptions, these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. Later, in [13], the authors proved that for q odd, the order of CD(n; q) is exactly 2q n Gammab n 2 4 c 1 . Combining this with a result from [9] on the existence of a girth cycle of length n 5 in D(n; q) for all odd n and infinitely many q, we get that the corresponding subfamily of graphs CD(n; q) ....

[Article contains additional citation context not shown here]

F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k; q), Discrete Mathematics 157 (1996) 271--283.

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F. Lazebnik, V. A. Ustimenko, A. J. Woldar, A characterization of the components of the graphs D(k; q), Discrete Math., 157 (1996), 271--283.

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F. Lazebnik, V.A. Ustimenko, and A.J. Woldar. A characterization of the components of the graphs D(k, q). Discrete Mathematics, 157(1-3):271--283, 1996. 22

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F. Lazebnik, V. Ustimenko, and A. Woldar. A characterization of the components of the graphs D(k; q). Discrete Mathematics, 157(1-3):271--283, 1996.

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F. Lazebnik, V.A. Ustimenko, and A.J. Woldar. A characterization of the components of the graphs D(k; q). Discrete Mathematics, 157(1-3):271--283, 1996.

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