3
votes
1answer
50 views

Show the relationship between the trace and the number of 4-cycles

Let $G$ be a k-regular graph. Show the exact relationship between $tr(A^4)$ and the number of 4-cycles in $G$. I understand how $tr(A^4)$ tells us the total number of closed paths of length 4 in ...
0
votes
0answers
26 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
2
votes
1answer
65 views

Algebraic Combinatorics

Let $K_{r,s}$ denote the complete bipartite graph, defined on $r + s$ vertices $\{v_1,v_2,...,v_r,w_1,...,w_s\}$, with an edge between $v_i$ and $w_j$ for $1 ≤ i ≤ r$ and $1 ≤ j ≤ s$. By ...
2
votes
0answers
51 views

Interesting Questions in Spectral Graph Theory

In the past, I have worked on few problems in Spectral graph theory and their applications to Physics. I have read parts of Fan Chung's book and Daniel Spielman lecture notes. I really enjoyed the ...
1
vote
1answer
88 views

Prove $MM^t=A+kI$ for matrices associated to graphs

How can I prove that $MM^t=A+kI$ for incidence matrix $M$ and adjacency matrix $A$ of a $k$-regular graph with $n$ vertices? It is easy to see that $MM^t$ is an $n\times n$-matrix (like $A$), ...
8
votes
1answer
177 views

Two formulas for the minimal eigenvalue of a graph

Hello again everybody, I'm reading Fan Chung's monograph Spectral Graph Theory. In it, she has two formulas for the minimal eigenvalue of a graph. She doesn't explain why they're equivalent, and I'm ...
12
votes
1answer
455 views

What does the minimal eigenvalue of a graph say about the graph's connectivity?

I'm reading Fan Chung's Spectral Graph Theory, and I'm now in chapter 2. There, Chung proves Cheeger's inequality, which is that $2h_G \geq \lambda_1 > h_G^2/2$ for any graph $G$. To me, this ...
2
votes
1answer
210 views

Justifying a pair of inequalities involving the exponential function

I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
0
votes
2answers
169 views

Question from section 1.5 of Chung's Spectral Graph Theory

I'm (slowly) reading Fan Chung's Spectral Graph Theory. At the moment, I'm in section 1.5 which is about eigenvalues and random walks. There's a small technical point that puzzles me. The context ...