Ordering starlike trees by the totality of their spectral moments

关于短词典序与带星状分支树的图的 $A_α$-谱半径之间的关系

On the relationship between shortlex order and $A_α$-spectral radii of graphs with starlike branch tree

本文研究在连通图顶点上添加星状分支路径所得图的 $A_α$-谱半径排序问题，证明该排序与对应整数分拆的短词典序完全一致，揭示了谱性质与组合序结构之间的内在联系。

On the largest real root of independence polynomials of graphs, an ordering on graphs, and starlike trees

Let $G$ be a simple graph of order $n$. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $I(G,x)=\sum_{k=0}^{n} s(G,k) x^{k}$, where $s(G,k)$ is the number of independent sets of $G$ of size $k$ and $s(G,0)=1$. Clearly all real roots of $I(G,x)$ are negative. Let $ξ(G)$ be the largest real root of $I(G,x)$. Let $H$ be a simple graph. By $G \succeq H$ we mean that $I(H,x)\geq I(G,x)$ for every $x$ in the interval $[ξ(G),0]$. We note that $G\succeq H$ implies that $ξ(G)\geq ξ(H)$. Also we let $G\succ H$ if and only if …

On the ordering of trees by the Laplacian coefficients

We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, \textit{Ordering trees by the Laplacian coefficients}, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For two $n$-vertex trees $T_1$ and $T_2$, if $c_k (T_1) \leqslant c_k (T_2)$ holds for all Laplacian coefficients $c_k$, $k = 0, 1, ..., n$, we say that $T_1$ is dominated by $T_2$ and write $T_1 \preceq_c T_2$. We proved that among $n$ vertex trees with fixed diameter $d$, the caterpillar $C_{n, d}$ has minimal Laplacian coefficients $c_k$, $k = 0, 1,..., n$. T…

The Harary index of trees

The Harary index of a graph $G$ is recently introduced topological index, defined on the reverse distance matrix as $H(G)=\sum_{u,v \in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path between two distinct vertices $u$ and $v$. We present the partial ordering of starlike trees based on the Harary index and we describe the trees with the second maximal and the second minimal Harary index. In this paper, we investigate the Harary index of trees with $k$ pendent vertices and determine the extremal trees with maximal Harary index. We also characterize the extremal trees wi…