On the Local Adjacency Metric Dimension of Generalized Petersen Graphs

<p class="Abstract">The local adjacency metric dimension is one of graph topic. Suppose there are three neighboring vertex , , in path . Path is called local if where each has representation: a is not equals and may equals to . Let’s say, . For an order set of vertices , the adjacency representation of with respect to is the ordered -tuple , where represents the adjacency distance . The distance defined by 0 if , 1 if adjacent with , and 2 if does not adjacent with . The set is a local adjacency resolving set of if for every two distinct vertices , and adjacent with y then . A minimum local adjacency resolving set in is called local adjacency metric basis. The cardinality of vertices in the basis is a local adjacency metric dimension of , denoted by . Next, we investigate the local adjacency metric dimension of generalized petersen graph.</p>

Infinite Excursions of Rotor Walks on Regular Trees

A rotor configuration on a graph contains in every vertex an infinite ordered sequence of rotors, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a rotor walk is a deterministic process: at each step it chooses the next unused rotor in its current location, and uses it to jump to the neighboring vertex to which it points. Rotor walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question for natural random configuration models on regular trees. Our results suggest that in this context the rotor model behaves like the simple random walk unless it is not "close to" the standard rotor-router model.

Foolproof eternal domination in the all-guards move model

The eternal domination problem requires a graph be protected against an infinitely long sequence of attacks at vertices, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. An attack is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow all guards to move to neighboring vertices in response to an attack, but allow the attacked vertex to choose which neighboring guard moves to the attacked vertex. This is the all-guards move version of the “foolproof” eternal domination problem that has been previously studied. We present some results and conjectures on this problem.