单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.

Improved Upper Bounds for the Largest Eigenvalue of Unicyclic Graphs

Let $G(V,E)$ be a unicyclic graph, $C_{m}$ be a cycle of length $m$ and $C_{m}\subset G$, and $u_{i}\in V(C_{m})$. The $G-E(C_{m})$ are $m$ trees, denoted by $T_{i}$, $i=1,2,\ldots,m$. For $i=1,2,\ldots,m$, let $e_{u_{i}}$ be the excentricity of $u_{i}$ in $T_{i}$ and $$e_{c}=\max\{e_{u_{i}}: i=1,2,\ldots,m\}.$$ Let $k=e_{c}$+1. For $j=1,2,\ldots,k-1$, let $$\delta_{ij}=\max\{d_{v}:{\rm dist}(v,u_{i})=j, v\in T_{i}\},$$ $$\delta_{j}=\max\{\delta_{ij}:i=1,2,\ldots,m\},$$ $$\delta_{0}=\max\{d_{u_{i}}:u_{i}\in V(C_{m})\}.$$ Then $$\lambda_{1}(G)\leq \max\{\max\limits_{2\leq j\leq k-2}(\sqrt{\delta_{j-1}-1}+\sqrt{\delta_{j}-1}),2+\sqrt{\delta_{0}-2},\sqrt{\delta_{0}-2}+\sqrt{\delta_{1}-1}\}.$$ If $G\cong C_{n}$, then the equality holds, where $\lambda_{1}(G)$ is the largest eigenvalue of the adjacency matrix of $G$.