All Questions

Let $G$ be a finite undirected graph. A closed walk in $G$ is a walk from any vertex of $G$ to itself. It is relatively straightforward to show that the total number of closed walks of length $k$ in $...

Given an irregular graph $G$ whose maximum degree is $d$, I am interested in producing a new graph $G'$ which is regular and has the spectrum of the adjacency spectrum of $G$ embedded in the spectrum ...

If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...

I am somewhat stuck working on an issue and would really love some guidance. I will state the problem, my current state and what led to it in case the solution lies beyond where I was looking The ...

Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...

The following question is from Introduction to dynamical systems, written by Michael Brin and Garrett Stuclk. Given two non-negative integer square matrices $A, B$, we say $A, B$ are elementarily ...