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Let G = (V (G),E(G)) be a simple strongly connected digraph and q(G) be the signless Laplacian spectral radius of G. For any vertex vi ∈ V (G), let d+i denote the outdegree of vi, m+i denote the average 2-outdegree of vi, and N+i denote the set of out-neighbors of vi. In this paper, we prove that: (1) (1) q(G) = d+1 +d+2 , (d+1 ≠ d+2) if and only if G is a star digraph [...] ,where d+1, d+2 are the maximum and the second maximum outdegree, respectively [...] is the digraph on n vertices obtained...

A graph $G$ is called $\{H_1,H_2,\cdots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\le i\le k$. We define $${\sigma }_k=min\left\{\sum _{i=1}^{k}d\left(v_i\right):\{v_1,\cdots ,v_k\}\text{is}\text{an}\text{independent}\text{set}\text{of}\text{vertices}\text{in}G\right\}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and ${\sigma }_3\left(G\right)\ge n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N\left(x_1\right)\cup N\left(x_2\right)|+|N\left(y_1\right)\cup N\left(y_2\right)|\ge n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.