Question

Find the
radius and
of
spectrom, spectral
spectral​

Answer 1

Step-by-step explanation:

The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.

This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C). In this case, for the graph G define:

{\displaystyle \ell ^{2}(G)=\left\{f:V(G)\to \mathbf {R} \ :\ \sum \nolimits _{v\in V(G)}\left\|f(v)^{2}\right\|<\infty \right\}.}\ell ^{2}(G)=\left\{f:V(G)\to {\mathbf {R}}\ :\ \sum \nolimits _{{v\in V(G)}}\left\|f(v)^{2}\right\|<\infty \right\}.

Let γ be the adjacency operator of G:

{\displaystyle {\begin{cases}\gamma :\ell ^{2}(G)\to \ell ^{2}(G)\\(\gamma f)(v)=\sum _{(u,v)\in E(G)}f(u)\end{cases}}}{\begin{cases}\gamma :\ell ^{2}(G)\to \ell ^{2}(G)\\(\gamma f)(v)=\sum _{{(u,v)\in E(G)}}f(u)\end{cases}}

The spectral radius of G is defined to be the spectral radius of the bounded linear operator γ.