lexicographic ordering by spectral moments ($s$-order) among all trees is discussed in this‎ ‎paper‎. ‎for two given positive integers $p$ and $q$ with $pleqslant q$‎, ‎we denote $mathscr{t}_n^{p‎, ‎q}={t‎: ‎t$ is a tree of order $n$ with a $(p‎, ‎q)$-bipartition}‎. furthermore, ‎the last four trees‎, ‎in the $s$-order‎, ‎among $mathscr{t}_n^{p‎, ‎q},(4leqslant pleqslant q)$ are characterized‎.

In order to have confidence in model-based phylogenetic methods, such as maximum likelihood (ML) and Bayesian analyses, one must use an appropriate model of molecular evolution identified using statistically rigorous criteria. Although model selection methods such as the likelihood ratio test and Akaike information criterion are widely used in the phylogenetic literature, model selection method...

A difference graph is a bipartite graph G = (X, Y: E) such that all the neighborhoods of the vertices of X are comparable by inclusion. We enumerate labeled and unlabeled difference graphs with or without a bipartition of the vertices into two stable sets. The labeled enumerations are expressed in terms of combinatorial numbers related to the Stirling numbers of the second kind.