Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic graph then q ≤ 2p− 3. Furthermore, if G is super edge-magic and q = 2p− 3, then the girth of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then q ≤ 2p − 5. In this paper we show that there are infinitely many graphs which are super edge-magic, have girth 5, and q =...

For every r ∈ N, let θr denote the graph with two vertices and r parallel edges. The θr-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θr. This notion generalizes the usual concept of girth which corresponds to the case r = 2. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213–225, 2003], Kühn and Osthus showed that gra...

Large-girth quasi-cyclic LDPC codes have been experimentally evaluated for use in PMD compensation by turbo equalization for a 10 Gb/s NRZ optical transmission system, and observing one sample per bit. Net effective coding gain improvement for girth-10, rate 0.906 code of length 11936 over maximum a posteriori probability (MAP) detector for differential group delay of 125 ps is 6.25 dB at BER o...

Objective: To determine the relationship between asymmetries in lower leg girth and standing heel-rise after anterior cruciate ligament (ACL) reconstruction. Design: Single-group posttest. Participants: 15 at a mean of 30 d after ACL reconstruction. Measurements: Lower leg girth and number of repetitions performed on the standing heel-rise test. Results: A significant decrease in lower leg girt...

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2-regular (following the notation of Djokovic), or...

The Moore bound m(d, k) = 1 + d ∑k−1 i=0 (d − 1)i is a lower bound for the number of vertices of a graph by given girth g = 2k + 1 and minimal degree d. Hoffmann and Singleton [5], Bannai and Ito [1], Damerell [4] showed that graphs with d > 2 tight to this bound can only exist for girth 5 and degree 3, 7, 57. The difference to the Moore bound by given girth is called the excess of a graph. In ...

Terwilliger [15] has given the diameter bound d < (s 1)(k 1) + 1 for distance-regular graphs with girth 2s and valency k. We show that the only distance-regular graphs with even girth which reach this bound are the hypercubes and the doubled Odd graphs. Also we improve this bound for bipartite distance-regular graphs. Weichsel [17] conjectures that the only distance-regular subgraphs of a hyper...

The girth is the length of shortest cycle in a graph. Recently, the problem for computing the girth of a graph have been studied well. Itai and Rodeh developed the first non trivial algorithm for computing girth in general graphs. Their algorithm runs in O (nm) time in the worst case and it runs in O(n) time on average case. For unweighted planer graphs, Djidjev gave a O ( n log n ) time algori...