Spectral integral variation and unicyclic 3-colored digraphs with second smallest eigenvalue 1

In this article, the spectral integral variation of weighted directed graphs occurring in one place by adding a weighted directed edge is studied. Our results extend those of So (1999) [13], for unweighted undirected graphs and Fan (2003) [6] for mixed graphs. We characterize the unicyclic 3-colored digraphs that have 1 as the second smallest Laplacian eigenvalue. Finally, as an application we characterize the unicyclic 3-colored digraphs for which spectral integral variation occurs in one place by adding an edge of color red and where the changed eigenvalue is the second smallest Laplacian eigenvalue.