论文标题
订购$ q $ - 图形：给定的大小和周长
Ordering $Q$-indices of graphs: given size and girth
论文作者
论文摘要
图谱理论中无标志性的拉普拉斯矩阵是图形的显着矩阵，研究人员对其进行了广泛的研究。 1981年，Cvetković指出了$ 12 $方向，以进一步研究图形光谱，其中之一是“分类和订购图”。除了这个经典的方向外，我们还要关注无标志性图形矩阵的最大特征值的顺序，该矩阵通常称为图形的$ q $ index。令$ \ mathbb {g}（m，g）$（分别$ \ mathbb {g}（m，\ geq g）$）为带有girth $ g $（resp。g$）的$ m $边缘上的连接图的家族，在$ g \ ge3 $的情况下。在本文中，我们首先订购第一个$（\ lfloor \ frac {g} {2} \ rfloor+2）$最大的$ q $ - $ q $ indices of $ \ mathbb {g}（m，g）$，其中$ m \ ge 3g ge 3g \ ge ge ge 12 $。其次，我们订购第一个$（\ lfloor \ frac {g} {2} \ rfloor+3）$最大$ q $ - $ q $ - $ \ mathbb {g}（m，\ geq g）$，其中$ m \ ge 3g ge 3g 3g \ ge ge 12 $。作为补充，我们将前五个最大的$ Q $ Q $ indices in $ \ mathbb {g}（m，3）$用$ m \ ge 9 $。最后，我们给出了所有连接图表的第一个最大$ q $ - Q $ Indices的订单。
The signless Laplacian matrix in graph spectra theory is a remarkable matrix of graphs, and it is extensively studied by researchers. In 1981, Cvetković pointed $12$ directions in further investigations of graph spectra, one of which is "classifying and ordering graphs". Along with this classic direction, we pay our attention on the order of the largest eigenvalue of the signless Laplacian matrix of graphs, which is usually called the $Q$-index of a graph. Let $\mathbb{G}(m, g)$ (resp. $\mathbb{G}(m, \geq g)$) be the family of connected graphs on $m$ edges with girth $g$ (resp. no less than $g$), where $g\ge3$. In this paper, we firstly order the first $(\lfloor\frac{g}{2}\rfloor+2)$ largest $Q$-indices of graphs in $\mathbb{G}(m, g)$, where $m\ge 3g\ge 12$. Secondly, we order the first $(\lfloor\frac{g}{2}\rfloor+3)$ largest $Q$-indices of graphs in $\mathbb{G}(m, \geq g)$, where $m\ge 3g\ge 12$. As a complement, we give the first five largest $Q$-indices of graphs in $\mathbb{G}(m, 3)$ with $m\ge 9$. Finally, we give the order of the first eleven largest $Q$-indices of all connected graphs with size $m$.
