On a family of trees with minimal atom-bond connectivity index

Zhibin Du, Carlos M. da Fonseca

Abstract: Let G=(V,E)<math><mi is="true">G</mi><mo is="true">=</mo><mrow is="true"><mo is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo is="true">)</mo></mrow></math> be a graph, di<math><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub></math> the degree of the vertex i<math><mi is="true">i</mi></math>, and ij<math><mi is="true">i</mi><mi is="true">j</mi></math> the edge incident to the vertices i<math><mi is="true">i</mi></math> and j<math><mi is="true">j</mi></math>. The atom-bond connectivity index (or, simply, ABC index) is defined as ABC(G)=∑ij∈E(di+dj−2)/(didj)<math><mi is="true">A</mi><mi is="true">B</mi><mi is="true">C</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo is="true">=</mo><msub is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">i</mi><mi is="true">j</mi><mo is="true">∈</mo><mi is="true">E</mi></mrow></msub><msqrt is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mo is="true">+</mo><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><mo is="true">−</mo><mn is="true">2</mn><mo is="true">)</mo></mrow><mo is="true">/</mo><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">j</mi></mrow></msub><mo is="true">)</mo></mrow></mrow></msqrt></math>. While this vertex-degree-based graph invariant is relatively well-known in chemistry, only recently a significant number of results emerged among the mathematical community. Though, several important problems remained open. One of them is the characterization of the tree(s) with minimal ABC index. In this paper, we will present some structural properties of one family of trees containing a pendent path of length 3 which would minimize the ABC index, mainly including: it contains no the so-called Bk<math><msub is="true"><mrow is="true"><mi is="true">B</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msub></math> with k⩾4<math><mi is="true">k</mi><mo is="true">⩾</mo><mn is="true">4</mn></math>, and contains at most two B2<math><msub is="true"><mrow is="true"><mi is="true">B</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>’s.