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The Perron-Frobenius theorem for nonnegative tensors gives us the symmetric property and stabilizing property of the spectrum of a weakly irreducible nonnegative tensor. The spectral symmetry is reflected by the cyclic index, which is defined to be the maximum positive integer $k$ such that the spectrum of the tensor keeps invariant under a rotation of angle $\frac{2\pi}{k}$ of the complex plane. The spectral stabilizing property is reflected by the stabilizing index, which is the order of a stablizlier group of the tensor. These two parameters are closely related to the the number of eigenvalues with modulus equal to spectral radius, and the number of eigenvectors of the tensor assocaited with the spectral radius, respectively. We give an explicit formula for the cyclic index by using the generalized traces, and proved that for any positive integer $k$ that divides $m$, there always exists an $m$-uniform hypergraph $G$ such that its adjacency tensor has the cyclic index $k$. If the tensor is further symmetric, then we can get the stabilizing index via the Smith normal form. We also show that for a weakly irreducible nonnegative tensor, there are finite many eigenvectors associated with the spectral radius up to a scalar.£¨Perron-Frobenius¶¨Àí¸ø³öÁË·Ç¸ºÈõ²»¿ÉÔ¼ÕÅÁ¿µÄÆ×¶Ô³ÆÐÔºÍÎÈ¶¨ÐÔ¡£Æ×¶Ô³ÆÐÔ¿ÉÓÉÑ»·Ö¸ÊýÀ´¿Ì»£¬¶¨ÒåÎª×î´óµÄÕýÕûÊýk, Ê¹µÃÕÅÁ¿µÄÆ×ÔÚ¸´Æ½ÃæÐý×ª$\frac{2\pi}{k}$ÏÂ±£³Ö²»±ä¡£Æ×ÎÈ¶¨ÐÔ¿ÉÓÉÎÈ¶¨Ö¸ÊýÀ´¿Ì»£¬¶¨ÒåÎªÕÅÁ¿µÄÎÈ¶¨×ÓµÄ½×¡£ÕâÁ½¸ö²ÎÊýÓëÄ£ÎªÆ×°ë¾¶µÄÌØÕ÷ÖµµÄÊýÄ¿£¬ÒÔ¼°ÓëÆ×°ë¾¶¹ØÁªµÄÌØÕ÷ÏòÁ¿µÄÊýÄ¿ÃÜÇÐÏà¹Ø¡£°ÄÃÅÄáÍþË¹ÈËÍøÕ¾8311Ó¦ÓÃ¹ãÒå¼£¸ø³öÑ»·Ö¸ÊýµÄÏÔÊ½¹«Ê½£¬Ö¤Ã÷ÁË¶ÔÈ°ÄÃÅÄáÍþË¹ÈËÍøÕ¾8311âÕû³ýmµÄÕýÕûÊýk, ×Ü´æÔÚÒ»¸öm-Ò»ÖÂ³¬Í¼G, Ê¹µÃGµÄÑ»·Ö¸ÊýÎªk. Èç¹ûÕÅÁ¿ÊÇ¶Ô³ÆµÄ£¬°ÄÃÅÄáÍþË¹ÈËÍøÕ¾8311Ó¦ÓÃSmith±ê×¼ÐÍ¸ø³öÎÈ¶¨Ö¸ÊýµÄÇó½â¡£°ÄÃÅÄáÍþË¹ÈËÍøÕ¾8311»¹Ö¤Ã÷ÁË·Ç¸ºÈõ²»¿ÉÔ¼ÕÅÁ¿µÄ¶ÔÓ¦ÓÚÆ×°ë¾¶µÄÌØÕ÷ÏòÁ¿ÊÇÓÐÏÞ¸ö¡££©

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