  We study a deterministic scale-free network recently proposed by
Barab\'{a}si, Ravasz and Vicsek. We find that there are two types of nodes: the
hub and rim nodes, which form a bipartite structure of the network. We first
derive the exact numbers $P(k)$ of nodes with degree $k$ for the hub and rim
nodes in each generation of the network, respectively. Using this, we obtain
the exact exponents of the distribution function $P(k)$ of nodes with $k$
degree in the asymptotic limit of $k \to \infty$. We show that the degree
distribution for the hub nodes exhibits the scale-free nature, $P(k) \propto
k^{-\gamma}$ with $\gamma = \ln3/\ln2 = 1.584962$, while the degree
distribution for the rim nodes is given by $P(k) \propto e^{-\gamma'k}$ with
$\gamma' = \ln(3/2) = 0.405465$. Second, we numerically as well as analytically
calculate the spectra of the adjacency matrix $A$ for representing topology of
the network. We also analytically obtain the exact number of degeneracy at each
eigenvalue in the network. The density of states (i.e., the distribution
function of eigenvalues) exhibits the fractal nature with respect to the
degeneracy. Third, we study the mathematical structure of the determinant of
the eigenequation for the adjacency matrix. Fourth, we study hidden symmetry,
zero modes and its index theorem in the deterministic scale-free network.
Finally, we study the nature of the maximum eigenvalue in the spectrum of the
deterministic scale-free network. We will prove several theorems for it, using
some mathematical theorems. Thus, we show that most of all important quantities
in the network theory can be analytically obtained in the deterministic
scale-free network model of Barab\'{a}si, Ravasz and Vicsek. Therefore, we may
call this network model the exactly solvable scale-free network.
