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...

Abstract Let G be a graph. Assume that to each vertex of set vertices $S\subseteq V(G)$ robot is assigned. At stage one can move neighbouring vertex. Then S mobile general position if there exists sequence moves the robots such all are visited while maintaining property at times. The number cardinality largest . We give bounds on and determine exact values for certain common classes graphs, inc...

In a connected graph G, the distance between two vertices of G is length shortest path these vertices. The eccentricity vertex u in largest and any other G. total-eccentricity index ?(G) sum eccentricities all this paper, we find extremal trees, unicyclic bicyclic graphs with respect to index. Moreover, conjugated trees

We define the swapping number of an arbitrary simple graph, which is related to edge reconstruction, and involves a weakening of the concept of a graph automorphism. We classify all 1-swappable trees and unicyclic graphs and prove that the expected value of the swapping number grows linearly with the order of the graph.

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, we determined the extremal (maximal and minimal) unicyclic and bicyclic graphs with respect to Harary index. 2010 Mathematics Subject Classification: 05C90