Our Soldiers and Lowest Classes of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their

vertices a very sharp and formidable angle.

Ten years ago Lavalle, "that imperturbable dreamer of the heavens," as Lazareff hailed him, gathered together the fruits of a lifetime's labour, and gave it, with well-justified contempt, to a world bound hand and foot to Barald's Theory of

Vertices and "compensating electric nodes." "They shall see," he wrote--in that immortal postscript to The Heart of the Cyclone--"the Laws whose existence they derided written in fire beneath them."

The number of

vertices in G is denoted by v(G), the number of its edges is denoted by e(G).

In social networks, the relations between character

vertices are tangled by the time difference of transaction, incompletion of personal information record, anonymous phenomena, and difference of information pattern and structure.

The distance between two

vertices u and v in a graph G, denoted by d(u, v),is defined as the length of a shortest path between u and v in G.

The Hamiltonian Cycle Problem (HCP) is to identify a cycle in an undirected graph connecting all the

vertices in the graph.

The lines in the famous Koch fractals [19] are mapped into

vertices, and there is an edge between two

vertices if two lines are connected; the generated novel networks were named Koch networks [20].

Let G = (V(G),E(G)) be a simple graph on n

vertices and m edges.

Some algorithms cannot effectively detect modifications in the topological relationship between

vertices; moreover, some algorithms require a high level of computational performance during the authentication code embedding procedure.

The open neighbourhood of a vertex v of G, denoted by [N.sub.G](v), is the set of

vertices adjacent to v.

A vertex-coloring S-edge-weighting of G is an assignment of weights by the elements of S to each edge of G so that adjacent

vertices have different sums of incident edges weights.