arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1907.04798math.CO
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双圈图的补图的最小无号Laplacian eigenvalue

The least signless Laplacian eigenvalue of the complements of bicyclic graphs

Xiaoyun Feng, Guoping Wang

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摘要：

设$G$是一个连通简单图，其顶点集为$V(G)=\{v_1, v_2,\cdots,v_n\}$ .则$G$的邻接矩阵是$A(G)=(a_{ij})_{n\times n}$，其中，如果$v_i$与$v_j$相邻，则为$a_{ij}=1$，否则为$a_{ij}=0$。度矩阵$D(G)=diag(d_{G}(v_1), d_{G}(v_2), \dots, d_{G}(v_n)),$，其中$d_{G}(v_i)$表示图$G$（$1\leq i\leq n$）中$v_i$的度。矩阵$Q(G)=D(G)+A(G)$称为$G$的无符号拉普拉斯矩阵。$Q(G)$的最小特征值也称为$G$的最小无符号拉普拉斯特征值。本文给出了两个嫁接变换，并利用它们刻画了在所有双圈图的补图中，最小无号Laplacian特征值最小的连通图。

英文摘要：

Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The degree matrix $D(G)=diag(d_{G}(v_1), d_{G}(v_2), \dots, d_{G}(v_n)),$ where $d_{G}(v_i)$ denotes the degree of $v_i$ in the graph $G$ ($1\leq i\leq n$). The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The least eigenvalue of $Q(G)$ is also called the least signless Laplacian eigenvalue of $G$. In this paper we give two graft transformations and then use them to characterize the unique connected graph whose least signless Laplacian eigenvalue is minimum among the complements of all bicyclic graphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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