查看更多>>摘要：For a simple graph G, let n(G), eta(G), r(G) and g(G) be respectively the order, the nullity, the rank and the girth of G. It was shown by Cheng and Liu (2007) that for every graph G, eta(G) <={n(G)- g(G)+ 2 if 4 vertical bar g(G) n(G)- g(G) if 4 inverted iota g(G). Connected graphs Gwith eta(G) = n(G)-g(G) +2 and n(G)- g(G) respectively have been characterized by Zhou etal. (2021). In this paper, we characterize connected graphs Gwith eta(G) = n(G)- g(G) - 1. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in Z). We also obtain sufficient conditions for the integrality of all distance powers of a Cayley graph over a given generalized dicyclic group. These results extend works on dicyclic groups by Cheng-Feng-Huang and Cheng-Feng-Liu-Lu-Stevanovic, respectively. (C) 2022 Elsevier Inc. All rights reserved.
Cameron, Thomas R.Hall, H. TracySmall, BenWiedemann, Alexander...
26页
查看更多>>摘要：In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. Notably, digraphs with a degenerate polygon (that is, a point or a line segment) as a restricted numerical range were completely described. In this article, we extend those results to include digraphs whose restricted numerical range is a non-degenerate convex polygon. In general, we refer to digraphs whose restricted numerical range is a degenerate or non-degenerate convex polygon as polygonal. We provide computational methods for identifying these polygonal digraphs and show that they can be broken into three disjoint classes: normal, restricted-normal, and pseudo-normal digraphs. Sufficient conditions for normal digraphs are provided, and we show that the directed join of two normal digraphs results in a restricted-normal digraph. Moreover, we prove that directed joins are the only restricted-normal digraphs when the order is square-free or twice a square-free number. Finally, we provide methods to construct restricted-normal digraphs that are not directed joins for all orders that are neither square-free nor twice a square-free number. (C) 2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier