C. T. Benson, Minimal regular graphs of girth eight and twelve, Canad. J. Math. 18 (1996), 1091-1094.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....bounds when lies between 2 and n. An easy upper bound is obtained as follows. For 3 let 2 = 2 (n) be the largest integer such that there are graphs G of order n, minimum degree at least 2 and girth at least 2 . The order of magnitude of 2 is only known for = 3; 4 and 6, see e.g. [2, 6]. So all facial cycles in a planar subgraph of such a graph G have length at least 2 and thus Euler s formula gives us an upper bound on the size of a planar subgraph of G. We believe that in general this upper bound is close to the truth (except maybe when the minimum degree is only a little ....

C. T. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math. 26 (1966), 1091-1094.

....2k )j 2 ckn (1 o(1) In Section 4 we include a sketch of the proof of Theorem 4. Note that for k = 2, 3 and 5, Theorem 4 is, apart from the factor c k in the exponent, best possible because, for these values of k, there are fC 4 ; C 2k g free graphs with Theta(n edges (see Benson [3] and Wenger [22] and 2 ex(n;fC4 ; C 2kg) is an obvious lower bound for jForb n (C 4 ; C 2k )j. For general k we remark that a conjecture of Erdos and Simonovits [10] states that ex(n; fC 4 ; C 2k g) Omega Gamma n ) It is crucial for the proof of Theorem 4 that the ....

C.T. Benson. Minimal regular graphs of girths eight and twelve. Canad. J. Math., 18:1091--1094, 1966.

....from the fact that Theorem 1 was tight. 2 Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r;s . Instead of Claim C, now we need . BS74] B66] [Be66], S66] W91] For a xed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satis es ex(n; G 2r ) O(n 1 1=r For r = 2; 3 and 5, this bound is tight. ....

C. Benson, Minimal regular graphs of girth eight and twelve, Canadian J. Mathematics 18 (1966), 1091-1094.

....from the fact that Theorem 1 was tight. # Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need . BS74] B66] [Be66], S66] W91] For a fixed positive integer r, let 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property 2r , satisfies 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. 00 . ....

C. Benson, Minimal regular graphs of girth eight and twelve, Canadian J. Mathematics 18 (1966), 1091--1094. 20

....k Delta (1 o(1) Delta m 2 4 k 2m (1 o(1) Delta 2 k Gamma8 k Delta m 1 2 k ; as desired. Constructions of graphs on m vertices with Omega Gamma m (k 1) k ) edges, which have no cycles of lengths 3; 2k, are known only for k = 2; 3; 5. In particular, Benson [Be 66] gave an algebraic construction, while Wenger [We 90] used finite projective geometries. Using the so called Ramanujan Graphs, Lubotzky, 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 ) ....

C. T. Benson, Minimal Regular Graphs of Girth Eight and Twelve, Canadian Journal of Mathematics 18, 1966, 1091-1094.

....the fact that Theorem 1 was tight. # Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . BS74] B66] [Be66], S66] W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. ....

C. Benson, Minimal regular graphs of girth eight and twelve, Canadian J. Mathematics 18 (1966), 1091--1094. 20

....infinity. The heat kernel of T k plays a central role in examining the spectrum of any kregular graph. To determine the heat kernel of T k , we can use the covering theorem in the previous section. The study of eigenvalues and eigenfunctions of T k can be found in many papers in the literature [1, 3, 9, 17, 19]. Here we will give a self contained proof for establishing the explicit formula for the kernel of the k tree, for k # 3. For thecaseofk=2,T 2 is just the infinite path. This special case and its cartesian products were examined in [6] T k can be regarded as a covering of the following weighted ....

C. T. Benson, Minimal regular graphs of girth eight and twelve, Canad. J. Math. 18 (1996), 1091-1094.

....the fact that Theorem 1 was tight. 2 Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . BS74] B66] [Be66], S66] W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. ....

C. Benson, Minimal regular graphs of girth eight and twelve, Canadian J. Mathematics 18 (1966), 1091--1094.

....infinity. The heat kernel of T k plays a central role in examining the spectrum of any kregular graph. To determine the heat kernel of T k , we can use the covering theorem in the previous section. The study of eigenvalues and eigenfunctions of T k can be found in many papers in the literature [1, 3, 9, 17, 19]. Here we will give a self contained proof for establishing the explicit formula for the kernel of the k tree, for k 3. For the case of k = 2, T 2 is just the infinite path. This special case and its cartesian products were examined in [6] T k can be regarded as a covering of the following ....

C. T. Benson, Minimal regular graphs of girth eight and twelve, Canad. J. Math. 18 (1996), 1091-1094.

....if and only if they correspond to an incident point line pair in the surface. The graphs Q q have order N = 2 q 4 Gamma1 q Gamma1 and degree Delta = q 1. For D = 6 the bipartite Moore graphs are called generalized hexagons, H q , and they are defined in a similar way to Q q , see [2]. They have order N = 2 q 6 Gamma1 q Gamma1 and degree Delta = q 1. Here follow some known results that will be used in the next Sections. If G is a ( Delta; D) bipartite graph and d(x; y) D, x; y 2 V (G) then there exist Delta disjoint paths between x and y of length exactly D. ....

.... Cayley graphs found by Campbell [6] CR Chordal rings found by Quisquater [16] vC Compound graphs designed by von Conta [8] Din Cayley graphs found by Dinneen [11] C n Cycle on n vertices GFS Special graph by G omez, Fiol and Serra [14] H q Incidence graph of a regular generalized hexagon [2] HS Hoffman Singleton graph K n Complete graph Lente Special graph designed by Lente, Univ. Paris Sud, France P Petersen graph P q Incidence graph of projective plane [15] Q q Incidence graph of a regular generalized quadrangle [2] T Tournament Operations G H twisted product of graphs ....

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Benson,C.T., Minimal regular graphs of girth eight and twelve, Canad. J. Math., 18(1966) 1091--1094.

....strong re 2 Girth LB Extra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte Coexter graph [18] 9 46 12 [22, 5] 8] 18 [8] 10 62 8 [1] 17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3 regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only simple fast tests at other places. 3. If there are choices to be made, select a decision with a minimum number of options. The savings in backtracking time can ....

....re 2 Girth LB Extra Found Opt. Cages 3 4 0 [18] K 4 [18] 4 6 0 [18] K 3;3 [18] 5 10 0 [18] Petersen graph [18] 6 14 0 [18] Heawood graph [18] 7 22 2 [12] McGee graph [19] 8 30 0 [18] Tutte Coexter graph [18] 9 46 12 [22, 5] 8] 18 [8] 10 62 8 [1] 17] 3 [21] 11 94 18 [2] NEW 12 126 0 [3] [3] Figure 1: The 3 regular cages of small girth dundancy tests near the root of the tree (where there are usually few nodes) but only simple fast tests at other places. 3. If there are choices to be made, select a decision with a minimum number of options. The savings in backtracking time can often ....

C. T. Benson. Minimal regular graphs of girths eight and twelve. Canadian J. of Math., 18:1091--1094, 1966.

....n 1 1=k (1 o(1) In Section 4 we include a sketch of the proof of Theorem 4. Note that for k = 2, 3 and 5, Theorem 4 is, apart from the factor c k in the exponent, best possible because, for these values of k, there are fC 4 ; C 2k g free graphs with (n 1 1=k ) edges (see Benson [3] and Wenger [22] and 2 ex(n;fC4 ; C 2k g) is an obvious lower bound for jForb n (C 4 ; C 2k )j. For general k we remark that a conjecture of Erd os and Simonovits [10] states that ex(n; fC 4 ; C 2k g) n 1 1=k ) It is crucial for the proof of Theorem 4 that the graphs ....

C.T. Benson. Minimal regular graphs of girths eight and twelve. Canad. J. Math., 18:1091-1094, 1966.

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C. T. Benson, Minimal regular graphs of girth eight and twelve, Canad. J. Math. 18 (1996), 1091-1094.

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C.T. Benson, Minimal regular graphs of girth eight and twelve, Canad. J. Math 18 (1966) 1091-1094.

No context found.

C.T. Benson. Minimal regular graphs of girth eight and twelve. Canadian Journal of Mathematics, 18:1091--1094, 1966.

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C. Benson. Minimal regular graphs of girth eight and twelve. Canadian Journal of Mathematics, 18:1091--1094, 1966.

No context found.

C.T. Benson. Minimal regular graphs of girth eight and twelve. Canadian Journal of Mathematics, 18:1091--1094, 1966.

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C. T. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math. 26 (1966), 1091-1094.

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C. T. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math. 26 (1966), 1091-1094.

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C. T. Benson, #Minimal regular graphs of girth eight and twelve," Canad. J. Math., 18 #1966#, 1091#1094.

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