论文标题
部分可观测时空混沌系统的无模型预测
On Zagreb index, signless Laplacian eigenvalues and signless Laplacian energy of a graph
论文作者
论文摘要
令$ g $是一个简单的图表，订单$ n $和尺寸$ m $。数量$ m_1（g）= \ displayStyle \ sum_ {i = 1}^{n} d^2_ {v_i} $称为$ g $的第一个Zagreb索引，其中$ d_ {v_i} $是$ v_i $的程度，对于所有$ v_i $，均为$ i = 1,2，2，2，2，2，2，dots，n $。图$ g $的无标志性拉普拉斯矩阵是$ q（g）= d（g）+a（g）$，其中$ a（g）$和$ d（g）$分别为$ g $的顶点级别的邻接和对角线矩阵。令$ q_1 \ geq q_2 \ geq \ dots \ geq q_n \ geq 0 $为$ g $的无符号laplacian特征值。最大的无标志性Laplacian特征值$ Q_1 $称为$ G $的无标志性Laplacian光谱半径或$ Q $ index，并用$ q（g）$表示。 Let $S^+_k(G)=\displaystyle\sum_{i=1}^{k}q_i$ and $L_k(G)=\displaystyle\sum_{i=0}^{k-1}q_{n-i}$, where $1\leq k\leq n$, respectively denote the sum of $k$ largest and smallest signless $ g $的laplacian特征值。 $ g $的无标志性拉普拉奇能量定义为$ qe（g）= \ displayStyle \ sum_ {i = 1}^{n} | q_i- \ overline {d} | $，其中$ \ overline {d} = \ frac {2m} {2m} {2m} {n} {n} {n} $是$ g $ $ g $。在本文中，我们获得了第一个Zagreb索引$ m_1（g）$的上限，并证明每个界限都是最好的。使用这些边界，我们为图形不变$ s^+_ k（g）$获得了几个上限，并表征了极端情况。结果，我们在各种图参数方面找到了$ q $ index的上限和图形不变$ l_k（g）$的下限，并确定极端情况。作为一种应用，我们获得了图形的无标志性拉普拉斯能量的上限，并表征了极端情况。
Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $A(G)$ and $D(G)$ denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of $G$. Let $q_1\geq q_2\geq \dots\geq q_n\geq 0$ be the signless Laplacian eigenvalues of $G$. The largest signless Laplacian eigenvalue $q_1$ is called the signless Laplacian spectral radius or $Q$-index of $G$ and is denoted by $q(G)$. Let $S^+_k(G)=\displaystyle\sum_{i=1}^{k}q_i$ and $L_k(G)=\displaystyle\sum_{i=0}^{k-1}q_{n-i}$, where $1\leq k\leq n$, respectively denote the sum of $k$ largest and smallest signless Laplacian eigenvalues of $G$. The signless Laplacian energy of $G$ is defined as $QE(G)=\displaystyle\sum_{i=1}^{n}|q_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average vertex degree of $G$. In this article, we obtain upper bounds for the first Zagreb index $M_1(G)$ and show that each bound is best possible. Using these bounds, we obtain several upper bounds for the graph invariant $S^+_k(G)$ and characterize the extremal cases. As a consequence, we find upper bounds for the $Q$-index and lower bounds for the graph invariant $L_k(G)$ in terms of various graph parameters and determine the extremal cases. As an application, we obtain upper bounds for the signless Laplacian energy of a graph and characterize the extremal cases.
