The vertex (resp. edge) metric dimension of a connected graph G is the size smallest set S?V(G) which distinguishes all pairs vertices edges) in G. In Sedlar and Škrekovski (2021) it was shown that both edge unicyclic always take values from just two explicitly given consecutive integers are derived structure graph. A natural problem arises to determine under what conditions these dimensions ea...

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the n-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian pe...

Zero forcing is an iterative graph coloring process whereby a colored vertex with a single uncolored neighbor forces that neighbor to be colored. It is NP-hard to find a minimum zero forcing set – a smallest set of initially colored vertices which forces the entire graph to be colored. We show that the problem remains NP-hard when the initially colored set induces a connected subgraph. We also ...

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n 2 ). In this paper we discuss the ...

Let G be a simple graph with vertex set V (G). A non-empty set S ⊆ V (G) is a global strong offensive alliance if for every vertex v in V (G)−S, a strict majority of its closed neighborhood is in S. The global strong offensive alliance number γô(G) is the minimum cardinality of a global strong offensive alliance of G. We show that if G is a connected bipartite graph of order at least three, the...

In the Metric Dimension problem, one asks for a minimum-size set R of vertices such that any pair graph, there is vertex from whose two distances to are distinct. This problem has mainly been studied on undirected graphs and gained lot attention in recent years. We focus directed graphs, show how solve linear-time digraphs underlying graph (ignoring multiple edges) tree. (nontrivially) extends ...