  The adjacency matrix is the most fundamental and intuitive object in graph
analysis that is useful not only mathematically but also for visualizing the
structures of graphs. Because the appearance of an adjacency matrix is
critically affected by the ordering of rows and columns, or vertex ordering,
statistical assessment of graphs together with their vertex sequences is
important in identifying the characteristic structures of graphs. In this
paper, we propose a hypothesis testing framework that assesses how locally
vertices are connected to each other along a specified vertex sequence, which
provides a statistical foundation for an optimization problem called envelope
reduction or minimum linear arrangement. The proposed tests are particularly
suitable for moderately small data and formulated based on a combinatorial
approach and a block model with intrinsic vertex ordering.
