Let G be a graph with n vertices and m edges. In many cases the complement of G has the following properties: it is connected, its diameter is 2, its Wiener index is equal to (n 2 ) +m, and its hyper-Wiener index is equal to (n 2 ) + 2m. We characterize the graphs whose complements have the mentioned properties.

Graovac and Pisanski [On the Wiener index of a graph, J. Math. Chem. 8 (1991) 53 – 62] applied an algebraic approach to generalize the Wiener index by symmetry group of the molecular graph under consideration. In this paper, exact formulas for this graph invariant under some graph operations are presented.

We derive the distribution of the center of mass S of the integrated superBrownian excursion (ISE) from the asymptotic distribution of the Wiener index for simple trees. Equivalently, this is the distribution of the integral of a Brownian snake. A recursion formula for the moments and asymptotics for moments and tail probabilities are derived.

We find the extremal values of the energy, the Wiener index and several vertex-degree-based topological indices over the set of Kragujevac trees with the central vertex of fixed degree. 2010 Mathematics Subject Classification : 05C90, 05C35.

In this note, we give expressions for the first(second) Zagreb coindex, second Zagreb index(coindex), third Zagreb index and first hyper-Zagreb index of the line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] and obtain upper bounds for Wiener index and degree-distance index of these graphs. This note continue the program of computing certain topological indi...

The edge-Wiener index of a connected graph is the sum of the distances between all pairs of edges of the graph. In this paper, we determine the polyphenyl chains with minimum and maximum edge-Wiener indices among all the polyphenyl chains with h hexagons. Moreover, explicit formulas for the edge-Wiener indices of extremal polyphenyl chains are obtained.

—We address the problem of linear minimum meansquared error (LMMSE) estimation under constraints on the lter or the estimated signal. We develop a general formula that leads to closed form solutions for a wide class of constrained LMMSE problems. The results are applicable to both nite dimensional problems as well as to the Wiener ltering setup, in which in nitely-many measurements are availabl...

Distance properties of molecular graphs form an important topic in chemical graph theory.1 To justify this statement just recall the famous Wiener index which is also known as the Wiener number. This index is the first2 but also one of the most important topological indices of chemical graphs. Its research is still very active; see recent reviews3,4 and several new results in a volume5 dedicate...