发表机构
Union University; University of Split; University of Ljubljana; Rudolfovo – Science and Technology Centre Novo Mesto; University in Novo Mesto（联合大学; 斯普利特大学; 卢布尔雅那大学; 鲁多尔福沃 - 诺沃梅斯托科技中心; 诺沃梅斯托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否证了Stanic关于连通二部图集合和树集合谱直径的两个猜想，分别给出了反例。
AI 中文摘要
设λ_1(G) ≥ λ_2(G) ≥ ... ≥ λ_n(G)为n个顶点图G的邻接谱。n顶点图G和H之间的谱距离σ(G,H)是它们谱之间的曼哈顿距离，即σ(G,H) = Σ_{i=1}^n |λ_i(G) - λ_i(H)|。给定一个由n阶两两非同构图组成的集合G，G的谱直径定义为sdiam(G) = max{secc_G(G): G ∈ G}，其中secc_G(G) = max{σ(G,H): H ∈ G, H与G不同构}是G在G中的谱离心率。在Stanic于2012年提出的关于谱距离的六个猜想中，有两个与某些图类的谱直径相关的猜想仍然开放。其中一个涉及所有n阶连通二部图集合B_n的谱直径，另一个涉及所有n阶树集合T_n的谱直径。更具体地说，Stanic猜想sdiam(B_n) = secc_Bn(K_{⌊n/2⌋,⌈n/2⌉})且sdiam(T_n) = σ(P_n, K_{1,n-1})。在本文中，这两个猜想都被否证了。
英文摘要
Let $λ_1(G) \geq \dots \geq λ_n(G)$ be the adjacency spectrum of a graph $G$ on $n$ vertices. The spectral distance $σ(G,H)$ between $n$-vertex graphs $G$ and $H$ is the Manhattan distance between their spectra, i.e. $σ(G,H) = \sum_{i=1}^n |λ_i(G) - λ_i(H)|$. Given a set $\mathcal{G}$ of pairwise non-isomorphic graphs of order $n$, the spectral diameter of $\mathcal{G}$ is defined as $\mathrm{sdiam}(\mathcal{G}) = \max\{\mathrm{secc}_{\mathcal{G}}(G) : G \in \mathcal{G}\}$, where $\mathrm{secc}_{\mathcal{G}}(G) = \max\{σ(G,H) : H \in \mathcal{G}\}$ is the spectral eccentricity of $G \in \mathcal{G}$. Among six conjectures on spectral distances posed by Z. Stanić in 2012, two conjectures related to the spectral diameter of certain graph classes remained open. One of them concerns the spectral diameter of the set $\mathcal{B}_n$ of all connected bipartite graphs of order $n$, while the other, of the set $\mathcal{T}_n$ of all trees of order $n$. More precisely, Stanić conjectured that $\mathrm{sdiam}(\mathcal{T}_n) = σ(P_n, K_{1,n-1})$, where $P_n$ is the path graph, while $K_{1,n-1}$ is the star, and that $\mathrm{sdiam}(\mathcal{B}_n) = \mathrm{secc}_{\mathcal{B}_n}(K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor})$, where $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$ is the complete bipartite graph. In this paper, both of these conjectures are disproved.
Comments15 pages, 2 figures
