On the maximal Randić energy of trees with given diameter

Document Type : Research Paper

Authors

1 Department of Mathematics, Shahed University, Tehran, Iran.

2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

Abstract

For given integers $n,d$ with $n\geq 5$ and $4\leq d \leq n-1$, let $T^{n}_d$ be the family of all trees of order $n$ and diameter $d$. In this paper, we study trees $T\in T^{n}_d$ with maximal Randić energy. We prove that if $T\in T^{n}_d$ is a tree with maximal Randić  energy then $T$ is obtained from a path $P=v_{0}v_{1} \ldots v_{d}$ by adding $ n_i$ path(s) $P_{3}$ to each vertex $v_{i}$, for $i= 2,3,4,\ldots,d-2$, where $n_i\in \{\lceil\frac{n-d+3}{2d-6}\rceil , \lfloor \frac{n-d+3}{2d-6}\rfloor\}$. In particular, we present families of trees satisfying the Gutman-Furtula-Bozkurt Conjecture proposed in [Linear Algebra Appl., 442 (2014), 50--57].

Keywords

References

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Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 14 November 2023

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