Furtula, "Topological index based on the ratios of geometrical and arithmetical means of

end-vertex degrees of edges," Journal of Mathematical Chemistry, vol.

The three cases (Point-Point, Point-Edge, and Point-Face) determine the distance equation [(8), (11), and (14)], and the two cases (max [D.sub.1] > max [D.sub.2] or max [D.sub.1] [less than or equal to] max [D.sub.2]) determine which point(s) can be moved: when max [D.sub.1] > max [D.sub.2], we update the location of the target

end-vertex (edge, face) on M' according to the Point-Point (PointEdge, Point-Face) case; when max [D.sub.1] [less than or equal to] max [D.sub.2], we update the locations of the end-vertices of the face containing the target point (i.e., the point achieving the greatest Hausdorff distance) on M'.

We now consider the case where all the links affected by a risk share an

end-vertex. Such failure scenario corresponds to risks like the cut of a conduit containing links issued from a node, or card failures in a router node.

If v [member of] V(G) and e [member of] E(G), we use v ~ e to denote that v is an

end-vertex of e.

Furtula, Topological index based on the ratios of geometrical and arithmetical means of

end-vertex degrees of edges, J.

On the other hand, the path [P.sub.2] shows that the lower bound in Theorem 9 can be sharp since [P.sub.2] has order n = 2 and diameter d = 1, while either

end-vertex of [P.sub.2] constitutes a distance pattern distinguishing set and so n + 1 - [2.sup.d] = 1 and [??]([P.sub.2]) = 1.

A vertex of degree one is termed an

end-vertex. In a [rho]-regular graph, [[rho].sub.i] has the same value [rho] for each vertex i.

If max{d(x),d(y)}[greater than or equal to]c/2 for any two vertices u and v with d(u,v)=2, then G has a longest path with at least an

end-vertex x satisfying d(x)[greater than or equal to]c/2 or G is a Hamiltonian.

The removal of an

end-vertex of the edges linking two cliques (e.g.

(c) H is nontraceable and has a 2-path cover [Q.sub.1], [Q.sub.2], such that [Q.sub.i] has an

end-vertex [a.sub.i] of degree 1 for i = 1,2, and all the end-vertices of [Q.sub.1] and [Q.sub.2] are independent.

Thus, an edge of Fab has one

end-vertex with zero at the first position and one

end-vertex with one at the first position.

Now, assume that deg(v) [greater than or equal to] 3 that means v is

end-vertex of a maximal simple path.