2022/06/30 by Wayne Barrett, Barrett, Wayne, Thomas R. Cameron +7
Chemistry · Computer Science · Mathematics · #Advanced Graph Theory Research #Graph theory and applications #Synthesis and Properties of Aromatic Compounds #math.CO #msc:05C20 #msc:05C50 #msc:15A18 #msc:15A60 #msc:52B20
paper · pdf · doi:10.48550/arxiv.2206.15410
arxiv created 2022/06/30 · arxiv updated 2022/07/01
In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between 0 and 1. Finally, we state several open conjectures that are motivated by empirical data.