  Lattice-based Cryptography is considered to have the characteristics of
classical computers and quantum attack resistance. We will design various
graphic lattices and matrix lattices based on knowledge of graph theory and
topological coding, since many problems of graph theory can be expressed or
illustrated by (colored) star-graphic lattices. A new pair of the
leaf-splitting operation and the leaf-coinciding operation will be introduced,
and we combine graph colorings and graph labellings to design particular proper
total colorings as tools to build up various graphic lattices, graph
homomorphism lattice, graphic group lattices and Topcode-matrix lattices.
Graphic group lattices and (directed) Topcode-matrix lattices enable us to
build up connections between traditional lattices and graphic lattices. We
present mathematical problems encountered in researching graphic lattices, some
problems are: Tree topological authentication, Decompose graphs into
Hanzi-graphs, Number String Decomposition Problem, $(p,s)$-gracefully total
numbers.
