In this paper, we mainly consider the first and second arc length variational problems on the exponential statistical manifold, and give the variational formulae. c ©2013 Mathematical Institute Slovak Academy of Sciences

For integers k, s with 0 ≤ k ≤ s ≤ |V (G)| − 3, a graph G is called s-Hamiltonian if the removal of any k vertices results in a Hamiltonian graph. For a simple connected graph that is not a path, a cycle or a K1,3 and an integer m ≥ 0, we define hs(G) = min{m : L(G) is s-Hamiltonian} and l(G) = max{m : G has an arc of length m that is not both of length 2 and in a K3}, where an arc in G is a pa...

A Helly circular-arc modelM = (C,A) is a circle C together with a Helly family A of arcs of C. If no arc is contained in any other, thenM is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, thenM is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal He...

Abstract. In a network the arc lengths may represent time or cost. In practical situations, it is reasonable to assume that each arc length is a type-2 discrete fuzzy set. We called it the type-2 discrete fuzzy shortest path problem. In this paper we proposed an algorithm for finding shortest path and shortest path length from source node to destination node using type reduction method. We have...

The standpoint is the station where the instrument is situated, and the forepoint is the point to which observations are being made (backsight or foresights). Figure 1, attached to this paper, shows the various distances discussed. They are described as follows: d1 is the wave path length (edm distance corrected for atmospheric delay), d2 is the wave path chord, d4 is the ellipsoidal arc length...

The standpoint is the station where the instrument is situated, and the forepoint is the point to which observations are being made (backsight or foresights). Figure 1, attached to this paper, shows the various distances discussed. They are described as follows: d1 is the wave path length (EDM distance corrected for atmospheric delay) d2 is the wave path chord d4 is the ellipsoidal arc length d...

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc D to a simplyconnected domain Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π, 4π), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most π in the special case in whi...

A 3 − arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ←→ G denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ←→ G . Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study t...

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is calle...