Adjacency spectrum, spectral radius, and cospectral graphs

Let $G$ be a finite simple graph, put $n:=\mathrm{\mid }V(G)\mathrm{\mid }$, and let $A(G)$ be an adjacency matrix of $G$ (The adjacency matrix of a finite simple graph).

Because $A(G)$ is real symmetric, the real spectral theorem gives a basis of real eigenvectors and shows that all roots of its characteristic polynomial are real (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, For $A\in M_n(F)$, the characteristic polynomial is ${\chi }_A(x)=\det (x{I}_n-A)$ when $n\ge 1$, with ${\chi }_A(x)=1$ for the unique $0\times 0$ matrix, For every finite-dimensional space, ${\sigma }_F(T)$ is exactly the set of roots in $F$ of ${\chi }_T$). If $n\ge 1$, we therefore list the eigenvalues in weakly decreasing order

${\lambda }_1(G)\ge {\lambda }_2(G)\ge \cdots \ge {\lambda }_n(G)\mathrm{.}$

If $n=0$, this list is empty. In either case, the multiset $\{{\lambda }_1(G),\dots ,{\lambda }_n(G)\}$, counted with multiplicities, is the adjacency spectrum of $G$.

The adjacency spectral radius of $G$ is $0$ when $n=0$, and otherwise is

$\rho (G):={\max }_{1\le i\le n}\mathrm{\mid }{\lambda }_i(G)\mathrm{\mid }\mathrm{.}$

Two finite graphs are cospectral when their adjacency spectra agree as multisets.