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These are usually known as the Laplacian, the normalized Laplacian and the unsigned Laplaian. All three are positive semidefinite. If the graph is regular, they all provide the same information. If t …

Suppose $G$ is a $k$-regular graph on $n$ vertices, with least eigenvalue $\tau$. Lovasz proved that $$ \theta(G) \le \frac{n}{1-\frac{k}{\tau}}. $$ Further if the automorphism group of $G$ acts …

Let $D$ be the Paley tournament on seven vertices. Its vertices are the integers mod seven and there is an arc from $i$ to $j$ is $j-i$ is a non-zero square mod seven. The characteristic polynomial of …

Since the eigenvalues are real, and since their sum is the trace of $A$, which is zero, we see that either all eigenvalues are zero, or there are both positive and negative eigenvalues. So no non-empt …

One approach that goes some way to explaining this is through the path-tree of a graph. This is defined as follows. Choose a vertex $u$ in the graph $G$, The vertices of the path-tree $T(G,u)$ are the …

Essentially your question is equivalent to asking for the relation between the spectrum of $A+D$ and $A$, where are $A$ is symmetric, $D$ is diagonal and both matrices are real. And, by change of basi …

Start with the unweighted case. We have $L=BB^T$ where $B$ is (what I call) the incidence matrix of an orientation of $G$. So $B$ has one 1 and one $-1$ in each column with all other entries zero. The …

(This is just an overlong comment.) A basic problem is that the complete bipartite graphs $K_{1,ab}$ and $K_{a,b}$ have the same spectral radius, and these graphs would not usually be viewed as simil …

We have $\det(tI-L) =\det(D^{-1}(tD-A))$. The matrix $D-A$ is positive semidefinite; it is the usual Laplacian in graph theory. The matrix $A+D$ is also positive semidefinite, and if the underlying gr …

Let $C$ be circulant of order $n\times n$. The first row of $C$ defines a complex function from the cyclic group $\mathbb{Z}_n$; denote its value on $i$ by $\rho(i)$. Define two elements $a$ and $b$ o …

The complete bipartite graphs $K_{n,n}$ are strongly regular and triangle-free. This nitpicking aside, your summary is accurate.

No. One simple class of examples are Latin square graphs. If $L$ is an $n\times n$ Latin square with entries from $\{1,\ldots,n\}$, the vertices of Latin square graph are the $n^2$ triples; two triple …

Let $M$ be a $d\times m$ matrix over $GF(2)$ and let $X(M)$ be the graph on the binary vectors of length $d$, where two vectors are adjacent in their difference is a column of $M$. (This is a Cayley g …

It you allow weighted adjacency matrices and if you insist (among there things) that the eigenspace associated to $\lambda_2$ satisfies the "strong Arnold condition", then you are dealing the Colin d …

View the vertices as elements of $\mathbb{Z}^n$. If $a\in\mathbb{Z}^n$, define a function $f_a$ on the vertices by $$ f_a(x) = (-1)^{a^Tx}. $$ This function is an eigenvectors and if $a$ has weight …