The concepts of covering and matching in fuzzy graphs using strong arcs are introduced and obtained the relationship between them analogous to Gallai’s results in graphs. The notion of paired domination in fuzzy graphs using strong arcs is also studied. The strong paired domination number γspr of complete fuzzy graph and complete bipartite fuzzy graph is determined and obtained bounds for the s...

the eccentricity connectivity index of a molecular graph g is defined as (g) = av(g)deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to othervertices of g and deg(a) is degree of vertex a. here, we compute this topological index forsome infinite classes of dendrimer graphs.

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the comp...

A graph G of order n is said to be borderenergetic if its energy is equal to 2n-2 and if G differs from the complete graph Kn. The first such graph was discovered in 2001, but their systematic study started only in 2015. Until now, the number of borderenergetic graphs of order n was determined for n

k-Interval Routing Scheme (k-IRS) is a compact routing method that allows up to k interval labels to be assigned to an arc. A fundamental problem is to characterize the networks that admit k-IRS. All of the problems related to single-shortest-path k-IRS have already been shown to be NP-complete. For all-shortest-path k-IRS, the characterization problems have been proved to be NP-complete for ev...

Given a weighted undirected graph G and a nonnegative integer k, the maximum k-star colorable subgraph problem consists of finding an induced subgraph of G which has maximum weight and can be star colored with at most k colors; a star coloring does not color adjacent nodes with the same color and avoids coloring any 4-path with exactly two colors. In this article, we investigate the polyhedral ...

One of the well-known NP-complete problems in graph theory is the Hamiltonian path problem; i.e., finding a simple path in the graph such that every vertex visits exactly once [5]. The two-dimensional integer grid G is an infinite undirected graph in which vertices are all points of the plane with integer coordinates and two vertices are connected by an edge if and only if the Euclidean distanc...

In 1857, the Irish mathematician Sir William Hamilton(1805-1865) invented a game of travelling around the edges of a graph from vertex to vertex. Hamilton described the game, in a letter to his friend Graves, as a mathematical game on the dodecahedron in which one person sticks five pins in any five consecutive vertices and the other is required to complete the path to form a spanning cycle. In...