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...

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥10 are injectively (...

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive graph. As an application of the result, we obtain some examples of weakly distanceregular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distanc...