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The atom-bond connectivity (ABC) index is one of the most actively studied degree-based graph invariants, which are found in a vast variety of chemical applications. For a simple graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, it is defined as <inline-formula> <tex-math notation="LaTeX">$ABC(G)=\sum _{uv \in E(G)} ({({d(u)+d(v)-2})/({d(u) d(v)})})^{1/2}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$d(v)$ </tex-math></inline-formula> denotes the degree of a vertex <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. Recently in <xref ref-type="bibr" rid="ref17">[17]</xref> graphs with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> vertices, <inline-formula> <tex-math notation="LaTeX">$2n-4$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$2n-3$ </tex-math></inline-formula> edges, and maximum <inline-formula> <tex-math notation="LaTeX">$ABC$ </tex-math></inline-formula> index were characterized. Here, we consider the next, more complex case, and characterize the graphs with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> vertices, <inline-formula> <tex-math notation="LaTeX">$2n-2$ </tex-math></inline-formula> edges, and maximum <inline-formula> <tex-math notation="LaTeX">$ABC$ </tex-math></inline-formula> index.

On the Maximum ABC Index of Graphs With Prescribed Size and Without Pendent Vertices