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A permutation \(\pi\) on a set of positive integers \(\{a_1, a_2, \ldots, a_n\}\) is said to be graphical if there exists a graph containing exactly \(a_i\) vertices of degree \(\pi(a_i)\) for each \(i\) (\(1 \leq i \leq n\)). It has been shown that for positive integers with \(a_1 < a_2 < \ldots < a_n\), if \(\pi(a_n) = a_n\), then the permutation \(\pi\) is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i)\).
We use a criterion of Tripathi and Vijay to provide a new proof of this result and to establish a similar result for permutations \(\pi\) such that \(\pi(a_{n-1}) = a_n\). We prove that such a permutation is graphical if and only if the sum \(\sum_{i=1}^n a_i \pi(a_i)\) is even and \(a_na_{n-1} \leq a_{n-1}(a_{n-1} – 1) + \sum_{i\neq n-1} a_i\pi(a_i)\). We also consider permutations such that \(\pi(a_n) = a_{n-1}\) and, more generally, those such that \(\pi(a_n) = a_{n-j}\) for some \(j\) (\(1 < j < n\)).