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A counterexample (to the unedited question): $$ A = \begin{pmatrix} 1 + x & 1 \\ 1 & 1 + x \end{pmatrix}. $$ Eigenvalues of $A$ are $2+x$ and $x$, principal minors have one eigenvalue $1+x$. Voting …

If graphs $X$ and $Y$ have adjacency matrices $A$ and $B$ respectively, then the composition of $X$ around $Y$ has adjacency matrix $$ A\otimes J + I\otimes B $$ Assume $B$ is $k$-regular. Then the …

Let $L(G)$ denote the Laplacian of $G$. Then $L(G)$ is positive semidefinite. If $\bar{G}$ denotes the complement of $G$ then $$ L(K_n) = L(\bar{G}) + L(G) $$ and so $L(K_n)-L(G)$ is positive semidef …

Since the Johnson graphs are regular, the Laplacian eigenvectors are the eigenvectors of the adjacency matrix. The Johnson graphs belong to an association scheme, the Johnson scheme, and explicit expr …

Let $D$ be a Steiner triple system on $v$ points. (So $v\equiv1,3$ mod 6). The incidence graph is the bipartite graph with the $v$ points as one colour class and the $v(v-1)/6$ blocks as the second; a …

If $Y$ is a 2-lift of $X$, there is a partition $\pi$ of $V(Y)$ into pairs, such that vertices in a pair are not adjacent and two distinct pairs are joined by a 2-matching, or by no edges at all. Assu …

We can do it in two steps. Step 1: show that if $A$ is a real symmetric matrix, there is an orthogonal matrix $L$ such that $A=LHL^T$, where $H$ is tridiagonal and its off-diagonal entries are non-n …

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 …

There's MR1186756: Godsil, C. D.; Hensel, A. D. Distance regular covers of the complete graph. J. Combin. Theory Ser. B 56 (1992), no. 2, 205–238. This only considers covers of complete graphs, but mu …

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 …