论文标题
具有最大程度的不规则图的光谱半径的最大值
Maxima of spectral radius of irregular graphs with given maximum degree
论文作者
论文摘要
令$λ^{*} $为具有最大度$δ$的$ n $顶点上连接的不规则图的最大光谱半径。 Liu，Shen和Wang（2007）指出，$ \ lim_ {n \ rightArrow \ infty}（n^{2}（n^{2}（δ-λ^{*}））/（δ-1）=π^{2} $，描述了不规则图的最大光谱范围的渐变性行为。为了关注这种猜想，我们考虑了连接的亚线二分组图的最大光谱半径。确定了具有最大光谱半径的唯一连接的亚采比二分图。令$ g $为$ k $连接的不规则图，具有频谱半径$λ_{1}（g）$，我们为$δ-λ_{1}（g）$提出了一个下限。此外，如果$ h $是$ k $连接的$δ$ -Gragrular Graph的适当子图，则还获得了$δ-λ_{1}（H）$的下限。这些界限改善了一些以前的结果。
Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a lower bound for $Δ-λ_{1}(G)$. Moreover, if $H$ is a proper subgraph of a $k$-connected $Δ$-regular graph, a lower bound for $Δ-λ_{1}(H)$ is also obtained. These bounds improve some previous results.
