The Inverse of the Incidence Matrix of a Unicyclic Graph

关于3色有向图的奇异性和逆矩阵

On the singularity and the inverse of 3-colored digraphs

本文研究了3色有向图的非奇异性和逆矩阵特性，探讨了非奇异单环和双环有向图的特征，并分析了其逆矩阵的零对角线条件及可逆性。

More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values…

有向图的复无符号拉普拉斯矩阵的摩尔 - 彭罗斯逆的组合公式

Combinatorial formula for the Moore-Penrose inverse of the complex signless Laplacian of an oriented graph

研究具有非零复边权重的弱连通有向图，通过找到无符号关联矩阵秩的充要条件，得出复无符号关联矩阵和复无符号拉普拉斯矩阵的摩尔 - 彭罗斯逆的组合公式，解决了相关开放问题。

On Sign-Invertible Graphs

Let $G$ be a graph and $A$ be its adjacency matrix. A graph $G$ is invertible if its adjacency matrix $A$ is invertible and the inverse of $G$ is a weighted graph with adjacency matrix $A^{-1}$. A signed graph $(G,σ)$ is a weighted graph with a special weight function $σ: E(G)\to \{-1,1\}$. A graph is sign-invertible (or sign-invertible) if its inverse is a signed graph. A sign-invertible graph is always unimodular. The inverses of graphs have interesting combinatorial interests. In this paper, we study inverses of graphs and provide a combinatorial description for sign-invertible graphs, whic…

有限域上点-球 incidences 的反向定理

Inverse Theorems for Point-Sphere Incidences over Finite Fields

该研究在有限域上证明了点-球 incidences 的反向定理，揭示了近极值性与代数刚性的关系，并解决了有限域上定点距离和点积问题的结构问题。

Incidence hypergraphs: The categorical inconsistency of set-systems and a characterization of quiver exponentials

This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e. quivers). The category of incidence hypergraphs is introduced as a "graph-like" remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is…

On the Inverse of Forward Adjacency Matrix

During routine state space circuit analysis of an arbitrarily connected set of nodes representing a lossless LC network, a matrix was formed that was observed to implicitly capture connectivity of the nodes in a graph similar to the conventional incidence matrix, but in a slightly different manner. This matrix has only 0, 1 or -1 as its elements. A sense of direction (of the graph formed by the nodes) is inherently encoded in the matrix because of the presence of -1. It differs from the incidence matrix because of leaving out the datum node from the matrix. Calling this matrix as forward adjac…

与梯图、环形梯图和莫比乌斯梯图相关的矩阵的显式公式

Explicit formulas for matrices associated to ladder, circular ladder, and Mobius ladder graphs

本文为梯图、环形梯图和莫比乌斯梯图推导出电阻距离矩阵及Moore-Penrose逆的显式公式，进而计算基尔霍夫指数并给出生成树数目的新组合公式。

图上的费米子与Zeta函数

Fermions and Zeta Function on the Graph

本文提出一种图上的费米子模型，其配分函数为图zeta函数的倒数，系数计数费米子圈数，并推广至网格图，证明无费米子加倍，可构造重叠费米子，且与统计模型及规范理论统一关联。

证明组合矩阵反演定理的局部框架

A local framework for proving combinatorial matrix inversion theorems

本文提出递归组合结构下矩阵互逆的局部化证明框架，并以 Kostka 矩阵、对称群特征标表等应用给出新的规范双射证明。