Moore-Penrose Inverses of the Signless Laplacian and Edge-Laplacian of Graphs

Incidence and Laplacian matrices of wheel graphs and their inverses

It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and an even unicyclic graph are quite different, we consider wheel graphs as they are formed from odd or even cycles. In this article solve the open problem for wheel graphs. This work has an interesting connection to inverses of circulant matrices.

Inverse formula for distance matrices of gear graphs

Distance matrices of some star like graphs are investigated in \cite{JAK}. These graphs are trees which are stars, wheel graphs, helm graphs and gear graphs. Except for gear graphs in the above list of star like graphs, there are precise formulas available in the literature to compute the inverse/Moore-Penrose inverse of their distance matrices. These formulas tell that if $D$ is the distance matrix of $G$, then $D^\dagger = -\frac{1}{2}L+uu'$, where $L$ is a Laplacian-like matrix which is positive semidefinite and all row sums equal to zero. The matrix $L$ and the vector $u$ depend only on th…

关于与某些图类相关的矩阵的广义逆

On generalized inverses of matrices associated with certain graph classes

本文研究双星有向图与D-连接星有向图相关矩阵的广义逆，给出了Drazin逆、群逆和Moore-Penrose逆的存在条件及显式公式，揭示了代数结构与图类组合性质的联系。

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。

The Inverse of the Incidence Matrix of a Unicyclic Graph

The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse of the incidence matrix of an even unicyclic graph is presented solving an open problem.

Application of Tikhonov Regularization in Generalized Inverse of Adjacency Matrix of Undirected Graph

In this paper, we found the Moore-Penrose generalized inverse of adjacency matrix of an undirected graph, explicitly. We proved that the matrix $R_λ= [r_{ij}]$ is nonsingular where $r_{ii}=\frac{1}λ+ °v_i$ and $r_{ij}=\mid N_G(V_i)\cap N_G(V_j)\mid$ for $i\neq j$, and we proved that $A^{\dagger}_G=[s_{ij}]_{1\leq i, j \leq n}$ where $\displaystyle{s_{ij}=s_{ji}=\lim_{λ\rightarrow +\infty} \langle R^{-1}_λe_j, f_i \rangle }$. The proof of the main result was based on the Tikhonov regularization.