雙分斯坦郝斯圖的計數

Abstract

斯坦郝斯矩陣為一對稱 $0-1$ 矩陣 $[a_{i,j}]_{n \times n}$，使得 當 $0 \leq i \leq n-1$ 時，$a_{i,i}=0$，且當 $1 \leq i < j \leq n-1$ 時， $a_{i,j}=(a_{i-1,j-1}+a_{i-1,j}) \pmod 2$。以斯坦郝斯 矩陣為相連矩陣所成的圖，稱做斯坦郝斯圖。在這篇論文中，我們給雙分 斯坦郝斯圖一個新的特徵，利用這個特徵，可求得雙分斯坦郝斯圖的個數 。 A Steinhaus matrix is a symmetric $0-1$ matrix $[a_{i,j}]_{n \times n}$ such that $a_{i,i}=0$ for $0 \leq i \leq n-1$ and $a_{i,j}=(a_{i-1,j-1}+a_{i-1,j}) \pmod 2$ for $1 \leq i<j \leq n-1$. A Steinhaus graph is a graph whose adjacency matrix is a Steinhaus matrix. In this paper, we present a new characterization of a graph to be a bipartite Steinhaus graph. From this characterization, we derive a fomula for the number $b(n)$ of bipartite Steinhaus graphs of order $n$.