  In this paper we study spectral properties of graphs which are constructed
from two given invertible graphs by bridging them over a bipartite graph. We
analyze the so-called HOMO-LUMO spectral gap which is the difference between
the smallest positive and largest negative eigenvalue of the adjacency matrix
of a graph. We investigate its dependence on the bridging bipartite graph and
we construct a mixed integer semidefinite program for maximization of the
HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive
upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of
semidefinite relaxation techniques. Several computational examples are also
presented in this paper.
