In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). A function f(p) is defined, and it is shown that γ×2(G) = min f(p), where the minimum is taken over the n-dimensional cube C = {p = (p1, . . ....

The values of the masses particles involved in decay ${T}_{cc}^{+}\ensuremath{\rightarrow}{D}^{*+}{D}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{D}^{0}{D}^{0}$ suggest that due to final state interactions transition vertex ${T}_{cc}^{+}\ensuremath{\rightarrow}{D}^{*+}{D}^{0}$ there may be triangle logarithmic singularities. We discuss their possible role and show tree approximation for c...

Consider two parallel lines (denoted r1 and r2). A graph is a PI graph (Point-Interval graph) if it is an intersection graph of a family F of triangles between r1 and r2 such that each triangle has an interval with two endpoints on r1 and a vertex (a point) on r2. The family F is the PI representation of G. The PI graphs are an extension of interval and permutation graphs and they form a subcla...

A pebbling move refers to the act of removing two pebbles from one vertex and placing pebble on an adjacent vertex. The goal graph is: Given initial distribution pebbles, use moves reach a specified called root. number \(\pi (G)\) is minimum needed so every can choice We introduce new variant pebbling, game between players. One player aims root other prevent this. show configurations various cl...

The edge versions of reverse Wiener indices were introduced by Mahmiani et al. very recently. In this paper, we find their relation with ordinary (vertex) Wiener index in some graphs. Also, we compute them for trees and TUC4C8(s) naotubes.

For a graph G, let 2(G) denote the minimum degree sum of a pair of nonadjacent vertices. Suppose G is a graph of order n. Enomoto and Ota (J. Graph Theory 34 (2000) 163–169) conjectured that, if a partition n = ∑k i=1 ai is given and 2(G)¿ n + k − 1, then for any k distinct vertices v1; : : : ; vk , G can be decomposed into vertex-disjoint paths P1; : : : ; Pk such that |V (Pi)| = ai and vi is ...

A positive integer n is said to be Wiener graphical, if there exists a graph G with Wiener index n. In this paper, we prove that any positive integer n( 6= 2, 5) is Wiener graphical. For any positive integer p, an interval [a, b] is said to be a p-Wiener interval if for each positive integer n ∈ [a, b] there exists a graph G on p vertices such that W (G) = n. For any positive integer p, an inte...

The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let Vn,k be the set of connected n-vertex graphs with connectivity at most k. In this note, we characterize the extremal (maximal and minimal) ...