On Modular-$2q$ Graphic Groups Of Topological Coding For Graphic Passwords

Since encrypting networks wholly by graphic groups, we will define a new labelling, called modular- $2q$ mixed odd-graceful labelling, and build up various graphic groups based on the modular- $2q$ mixed odd-graceful labelling. We will show: the necessity and sufficiency of subgroup of a graphic group; a ( $p, q$ ) -graph $G$ admitting a modular- $2q$ mixed(negative) odd-graceful labelling can generate a graphic group, and moreover these graphs form an Abelian additive graphic group. A ( $p, q$ ) -graph $G$ admits a modular- $2q$ odd-graceful labelling $f$ generate $2q$ vertex-modular graphs, each of them generates $2q$ edge-modular graphs, which form an Abelian additive graphic group, similar results can be obtained for modular- $2q$ negative odd-graceful graph and modular- $2q$ mixed odd-graceful graph.