Augmented nodal matrices and normal trees

Augmented nodal matrices play an important role in the analysis of different features of electrical circuit models. Their study can be addressed in an abstract setting involving two- and three-colour weighted digraphs. By means of a detailed characterization of the structure of proper and normal trees, we provide a unifying framework for the rank analysis of augmented matrices. This covers in particular Maxwell’s tree-based determinantal expansions of (non-augmented) nodal matrices, which can be considered as a one-colour version of our results. Via different colour assignments to circuit devices, we tackle the DC-solvability problem and the index characterization of certain differential-algebraic models which arise in the nodal analysis of electrical circuits, extending several known results of passive circuits to the non-passive context.     

A Note on the Order of Iterated Line Digraphs

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices of the k‐iterated line digraph , for , where . We obtain this result by using the minimal polynomial of a quotient digraph of G.     

Imprimitive non-powerful sign pattern matrices with maximum base

You et al. [L.H. You, J.Y. Shao, and H.Y. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007), pp. 285–300], obtained an upper bound of the bases for imprimitive non-powerful sign pattern matrices. In this article, we characterize those imprimitive non-powerful sign pattern matrices whose bases reach this upper bound.     

The kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs

In this article, we study the kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs. A bound on the kth upper bases for primitive nonpowerful minimally strong signed digraphs is obtained, and the equality case of the bound is characterized. For the kth lower bases, we obtain some bounds. For some cases, the bounds are best possible and the extremal signed digraphs are characterized. We also show that there exist ‘gaps’ in both the kth upper base set and the kth lower base set of primitive nonpowerful minimally strong signed digraphs.     

基于串并联电阻法的半偏思想测量电表内阻

利用中学物理实验中半偏法测量电表内阻的思想,通过给电表并联和串联电阻,调节串并联电阻的阻值使电表指针达到满偏和半偏的方法测量电表内阻,拓展了测量电表内阻的方法。     

On DP-coloring of digraphs

DP-coloring is a relatively new coloring concept by Dvořák and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\text{◂{}▸}\{\left(v,c\right)|\text{◂+▸}v\in V\left(G\right),\text{◂+▸}c\in L\left(v\right)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks’ type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.