For all positive even integers $n$, graphs of order $n$ with degree sequence \begin{ equation*} S_{ n}:1,2,\dots,n/2,n/2,n/2+1,n/2+2,\dots,n-1 \end{ equation*} naturally arose in the study of a labeling problem in \cite{ IMO}. This fact motivated the authors of the aforementioned paper to study these sequences and as a result of this study they proved that there is a unique graph of order $n$ realizing $S_{ n}$ for every even integer $n$. The main goal of this paper is to generalize this result.
Categories: math.CO