关于具有圈亏或圈过剩的图
On graphs with cyclic defect or excess
摘要：
Moore界构成最大度$d$和直径$D=k$的图的阶的上界和最小度$d$和奇围长$g=2k+1$的图的阶的下界。缺少或超过$ε$的摩尔界限的图分别被称为{ \it graphs with defect or excess $ε$ }。虽然{ \it摩尔图}（具有$ε=0$的图）和具有缺陷或过量1的图几乎已经完全表征，但具有缺陷或过量2的图代表了广泛的未探索区域。缺陷（过量）为2的图满足等式$G_{d,k}(A) = J_n + B$（$G_{d,k}(A) = J_n-B$），其中$A$表示所讨论的图的邻接矩阵，$n$表示其阶，$J_n$表示其条目都是1的$n\times n$矩阵，$B$表示顶点不相交循环的并集的邻接矩阵，以及$G_{d,k}(x)$是具有整数系数的多项式，使得矩阵$G_{d,k}(A)$给出连接所述矩阵中的每对顶点的长度至多为$k$的路径的数目。graph.特别是，如果$B$是一个循环的顺序$n$的邻接矩阵，我们称之为相应的图\emph{graphs with cyclic defect or excess}，这些图是我们在本文中关注的主题。我们证明了不存在无穷多个这样的图。作为本文的重点，我们给出了具有奇数度$d\ge3$和循环亏损或循环过剩的图的个数的$O(\frac{64}3d^{3/2})$的渐近上界.这个界实际上是相当慷慨的，作为一种说明的方式，我们证明了不存在的一些家庭的图的奇数度$d\ge3$和循环亏损或过剩。实际上，我们猜想，除了8个顶点上的莫比乌斯阶梯，没有任何程度的非平凡图$\ge 3$和循环亏损或过剩存在。
英文摘要：
The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $ε$ are called {\it graphs with defect or excess $ε$}, respectively. While {\it Moore graphs} (graphs with $ε=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n-B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs \emph{graphs with cyclic defect or excess}; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.
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