Some forbidden combinations of branches in minimal-ABC trees

The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑uv∈Ed(u)+d(v)−2d(u)d(v), where d(u) is the degree of vertex u in G and E denotes the set...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2018-02, Vol.236, p.165-182
Main Authors: Dimitrov, Darko, Du, Zhibin, da Fonseca, Carlos M.
Format: Article
Language:English
Subjects:
Atom-bond connectivity index
Atoms & subatomic particles
Chemicals
Extremal graphs
Graph theory
Tree
Trees (mathematics)
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Summary:The atom-bond connectivity (ABC) index has been, in recent years, one of the most actively studied vertex-degree-based graph invariants in chemical graph theory. For a given graph G, the ABC index is defined as ∑uv∈Ed(u)+d(v)−2d(u)d(v), where d(u) is the degree of vertex u in G and E denotes the set of edges of G. In this paper, we present some new structural properties of trees with a minimal ABC index (also refer to as a minimal-ABC tree), which is a step further towards understanding their complete characterization. We show that a minimal-ABC tree cannot simultaneously contain a B4-branch and B1 or B2-branches.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2017.11.003