New family of small regular graphs of girth 5

Abstract A ( k , g ) -cage is a k-regular graph of girth g of minimum order. In this work, we focus on girth g = 5 , where cages are known only for degrees k ≤ 7 . When k ≥ 8 , except perhaps for k = 57 , the order of a ( k , 5 ) -cage is strictly greater than 1 + k 2 . Considering the relationship between finite geometries and graphs we establish upper constructive bounds that improve the best so far.

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