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For a connected graph G = (V, E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is g ev (G). Any edge-to-vertex geodetic set of cardinality g ev (G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S is called a forcing subset for S if S is the unique minimum edge-to-vertex geodetic set containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of S, denoted by f ev (S), is the cardinality of a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of G, denoted by f ev (G), is f ev (G) = min {f ev (S)}, where the minimum is taken over all minimum edgeto-vertex geodetic sets S in G. Some general properties satisfied by the concept forcing edge-to-vertex geodetic number is studied. The forcing edge-to-vertex geodetic number of certain classes of graphs are determined. It is shown that

Let G = (V, E) be a graph. A subset S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is gev(G). Any edge-to-vertex geodetic set of cardinality gev(G) is called an edge-to-vertex geodetic basis of G. A connected edge-to-vertex geodetic set of a graph G is an edge-to-vertex geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge-to-vertex geodetic set of G is the connected edge-to-vertex geodetic number of G and is denoted by gcev(G). Some general properties satisfied by this concept are studied. The connected graphs G of size q with connected edge-to-vertex geodetic number 2 or q or q − 1 are characterized. It is shown that for any three positive integers q, a and b with 2 ≤ a ≤ b ≤ q, there exists a connected graph G of size q, gev(G) = a and gcev(G) = b.

A connected graph G is said to be super connected if every minimum vertex-cut isolates a vertex of G. Moreover a graph is said to be hyper connected if for every minimum vertex cut S, G − S has exactly two components, one of which is an isolated vertex. In this paper, we focussed the concept in arithmetic graphs V n and proved that, for an arithmetic graph G = V n , n = P a 1 1 × P a 2 2 × .... × P a r r where 0 < a i ≤ 2 and r > 3 is super and hyper connected and for at least one a i ≥ 3, the graph G = V n is only super connected. Also, it is clear that for every arithmetic graph G = V n , n is any integer is super and hyper edge connected.

Let G = (V, E) be a simple connected graph. The eccentric-distance sum of G is defined as ξ ds (G) = u∈V (G) e(u)D(u) where e(u) is the eccentricity of the vertex u in G and D(u) is the sum of distances between u and all other vertices of G. In this paper, we establish formulae to calculate the eccentric-distance sum for some cycle related graphs, namely Cn, complement of Cn, shadow of Cn and the line graph of Cn. Also, it is shown that, the eccentric-distance sum of Cn is less than the eccentric-distance sum of shadow of Cn for all n ≥ 3.