Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2021-07-05 , DOI: 10.1080/03081087.2021.1948956 Debajit Kalita 1 , Kuldeep Sarma 1
This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight ±i and with a unique perfect matching, denoted by , is considered in this article. We characterize the 3-coloured digraphs in whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight ±i satisfying the strong reciprocal eigenvalue property.
中文翻译:
关于单环三色有向图的逆
本文提供了非奇异三色有向图的邻接矩阵逆矩阵的组合描述。循环权重为±i且具有唯一完美匹配的单环三色有向图类,表示为, 在本文中被考虑。我们将 3 色有向图的特征描述为其逆又是三色有向图。此外,中的 3 色有向图其逆是二分的也有特征。证明了中的三色有向图的逆总是拉普拉斯非奇异的。中单环三色有向图的特征本文还提供了拥有单环逆元。作为一个应用,我们可以获得循环权重±i满足强倒数特征值性质的单环三色有向图类。