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spectra of graphs; spectral radius; $\ddag $-shape tree
Let $A(G)$ be the adjacency matrix of $G$. The characteristic polynomial of the adjacency matrix $A$ is called the characteristic polynomial of the graph $G$ and is denoted by $\phi (G, \lambda )$ or simply $\phi (G)$. The spectrum of $G$ consists of the roots (together with their multiplicities) $\lambda _1(G)\geq \lambda _2(G)\geq \ldots \geq \lambda _n(G)$ of the equation $\phi (G, \lambda )=0$. The largest root $\lambda _1(G)$ is referred to as the spectral radius of $G$. A $\ddag $-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by $G(l_1, l_2, \ldots , l_7)$ $(l_1\geq 0$, $l_i\geq 1$, $i=2,3,\ldots , 7)$ a $\ddag $-shape tree such that $G(l_1, l_2, \ldots , l_7)-u-v=P_{l_1}\cup P_{l_2}\cup \ldots \cup P_{l_7}$, where $u$ and $v$ are the vertices of degree 4. In this paper we prove that $3\sqrt {2}/{2}< \lambda _1(G(l_1, l_2, \ldots , l_7))< {5}/{2}$.
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