Maximum nullity and zero forcing number on graphs with maximum degree at most three

Meysam Alishahi, Elahe Rezaei-Sani, Elahe Sharifi

Abstract: A dynamic coloring of a graph G<math><mi is="true">G</mi></math> starts with an initial subset F⊆V(G)<math><mrow is="true"><mi is="true">F</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">⊆</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math> of colored vertices, while all the remaining vertices are non-colored. At each time step, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set F<math><mi is="true">F</mi></math> is called a zero forcing set of G<math><mi is="true">G</mi></math> if, by iteratively applying the forcing process, every vertex in G<math><mi is="true">G</mi></math> becomes colored. The zero forcing number of G<math><mi is="true">G</mi></math>, denoted by F(G)<math><mrow is="true"><mi is="true">F</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math>, is the cardinality of a minimum zero forcing set of G<math><mi is="true">G</mi></math>. The maximum nullity of G<math><mi is="true">G</mi></math>, denoted by M(G)<math><mrow is="true"><mi is="true">M</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math>, is the largest possible nullity over all |V(G)|<math><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo is="true">|</mo></mrow></math> by |V(G)|<math><mrow is="true"><mo is="true">|</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo is="true">|</mo></mrow></math> real symmetric matrices A<math><mi is="true">A</mi></math> whose non-diagonal entries are non-zero if the corresponding vertices are adjacent in G<math><mi is="true">G</mi></math> and with no restriction for its diagonal entries. In this paper, we characterize all graphs G<math><mi is="true">G</mi></math> of order n<math><mi is="true">n</mi></math>, maximum degree at most three, and F(G)=3<math><mrow is="true"><mi is="true">F</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">3</mn></mrow></math>. Also we classify these graphs with their maximum nullity.