Abstract
Let \(G\) be a simple and connected graph. A vertex \(v_i\) is said to be pendent if \(d_G(v_i)=1\), and its adjacent vertex is called a quasi-pendent vertex. Let \(\mathcal{T}(n)\) be a set of trees of order \(n\) with at most two quasi-pendent vertices
of degree less than \(4\). We name \(\mathcal{T}(n)\) the set of coral trees. The Randi\'{c} matrix of \(G\), denoted by \(R(G)\), is an \(n\times n\) matrix whose \((i,j)\)-entry is equal to \(\frac{1}{\sqrt{d_{G}(v_{i})d_{G}(v_{j})}}\)
if \(v_iv_j\in E(G)\), and \(0\) otherwise. The Randi\'{c} energy of \(G\) is defined as \[ {\cal E}_{R}(G)=\sum\limits^{n}_{i=1}|\mu_{i}(G)|, \] where \(\mu_{1}(G),\mu_{2}(G),\dots,\mu_{n}(G)\) are the eigenvalues of \(R(G)\). In [I. Gutman, B. Furtula, S. B. Bozkurt, On Randić energy, Linear Algebra Appl. 442 (2014) 50--57],
the authors conjectured that for a tree \(T\) of order \(n\), if \(n\) is odd, then the maximum \({\cal E}_{R}(T)\) is achieved for \(T\) being the (\(\frac{n-1}{2}\))-sun; if \(n\) is even, then the maximum \({\cal E}_{R}(T)\) is achieved
for \(T\) being the (\(\lceil \frac{n-2}{4} \rceil, \lfloor \frac{n-2}{4}\rfloor\))-double sun. In this work, we get the following results. (1) For \(T\in\mathcal{T}(n)\), \({\cal E}_{R}(T)\le{\cal E}_{R}(P_{n}).\) (2) For a graph \(G\),
if \({\cal E}_{R}(G)\le{\cal E}_{R}(P_{n}),\) then \(G\) satisfies the conjecture. (3) \(\mathcal{T}(n)\) is a family of trees that satisfies the conjecture.
