A stability result for girth‐regular graphs with even girth

Abstract

Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth-cycles containing e. For a vertex v of Γ let {e 1 , e 2 , . . . , e k } be the set of edges incident to v orderd such that n(e 1 ) ≤ n(e 2 ) ≤ • • • ≤ n(e k ). Then (n(e 1 ), n(e 2 ), . . . , n(e k )) is called the signature of v. The graph Γ is said to be girth-regular, if all of its vertices have the same signature.Let Γ be a girth-regular graph with girth g = 2d and signature (a 1 , a 2 , . . . , a k ) . It is known that in this case we have a k ≤ (k − 1) d . In this paper we show that if a k = (k − 1) d − for some non-negative integer < k − 1, then = 0.

On girth-biregular graphs

“…The aim of the present paper is to extend some of the results of [9] to the bipartite biregular case. If the valencies in the bipartition classes are k 1 > k 2 > 2, then we prove that the maximum number of girth-cycles containg an edge is at most M = (k 1 − 1) ⌊g/4⌋ (k 2 − 1) ⌈g/4⌉ , see Theorem 2.6.…”

“…Therefore, Γ is in fact girth-regular graph. As girth regular graphs were studied in details in [9] and [14], we will assume k 1 > k 2 for the rest of this paper.…”

“…The paper by Jajcay, Kiss and Miklavič [8] defined a new type of regularity: a graph is called edge-girth regular if the number of cycles of length g (the girth) containing an edge is independent of the edge. This definition was weakened by Potočnik and Vidali [14] and in [9] it was extended to a stability theorem. One can introduce the signature (a 1 , .…”

“…For such graphs with valency k ≥ 3, it was shown in [14] that a k ≤ (k − 1) 2d , where d = ⌊g/2⌋. In [9], the upper bound was improved for g = 2d in the sense that it is either (k − 1) 2d or at most (k − 1) 2d − (k − 1). In the former case the graph has to be the incidence graph of a thick generalized d-gon of order (k − 1, k − 1).…”