On the Aα-characteristic polynomial of a graph
On the Aα-characteristic polynomial of a graph
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关于图的 Aα 特征多项式
DOI：
10.1016/j.laa.2018.02.014
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发表时间：
2018
影响因子：
1.1
通讯作者：
Shunyi Liu
中科院分区：
文献类型：
--
作者：
Xiaogang Liu;Shunyi Liu
设G是一个有n个顶点的图，设A(G)和D(G)分别表示G的邻接矩阵和度矩阵。对于任意实数α∈[0, 1]，定义A α (G)= α D (G)+(1− α) A (G)。 G 的 A α 特征多项式定义为 det (x I n− A α (G))=Σ j c α j (G) x n− j，其中 det (⁎) 表示⁎的行列式，I n 是大小为 n 的单位矩阵。 G 的 A α 谱由 G 的 A α 特征多项式的所有根组成。如果所有与 G 具有相同 A α 谱的图都同构于 G，则称图 G 由其 A α 谱确定。在本文中，我们首先制定了前四个系数 c α 0 (G)、c α 1 (G)、c α 2 (G) 和 c α 3 (G) G 的 A α 特征多项式。然后，通过枚举最多 10 个顶点上的所有图的 A α 特征多项式，我们观察到 A α 谱对于我们区分图非常有效。为了验证这一观察结果，我们对一些由 A α 谱确定的图进行了表征。
Let G be a graph with n vertices, and let A (G) and D (G) denote respectively the adjacency matrix and the degree matrix of G. Define A α (G)= α D (G)+(1− α) A (G) for any real α∈[0, 1]. The A α-characteristic polynomial of G is defined to be det (x I n− A α (G))=∑ j c α j (G) x n− j, where det (⁎) denotes the determinant of⁎, and I n is the identity matrix of size n. The A α-spectrum of G consists of all roots of the A α-characteristic polynomial of G. A graph G is said to be determined by its A α-spectrum if all graphs having the same A α-spectrum as G are isomorphic to G. In this paper, we first formulate the first four coefficients c α 0 (G), c α 1 (G), c α 2 (G) and c α 3 (G) of the A α-characteristic polynomial of G. And then, we observe that A α-spectra are much efficient for us to distinguish graphs, by enumerating the A α-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their A α-spectra.