Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs

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

Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs

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 $v_i \in V (G)$, let $d+i$ denote the outdegree of $v_i$, $m_i^+$ denote the average 2-outdegree of $v_i$, and $N_i^+$ denote the set of out-neighbors of $v_i$. In this paper, we prove that: (1) $q(G) = d_1^+ + d_2^+, (d_1^+ \ne d_2^+ )$ if and only if $G$ is a star digraph $\overleftrightarrow{K}_{1,n-1}$, where $d_1^+$, $d_2^+$ are the maximum and the second maximum outdegree, respectively ($\overleftrightarrow{K}_{1,n-1}$ is the digraph on $n$ vertices obtained from a star graph $K_{1,n−1}$ by replacing each edge with a pair of oppositely directed arcs). (2) $q(G) \le \text{max} \bigg\{ \frac{1}{2} \left( d_i^+ + \sqrt{ { d_i^+ }^2 + 8d_i^+ m_i^+ } \right) : v_i \in V(G) \bigg\}$ with equality if and only if $G$ is a regular digraph. (3) $q(G) \le \text{max} \bigg\{ \frac{1}{2} \left( d_i^+ + \sqrt{ {d_i^+}^2 + \frac{4}{d_i^+} \sum_{v_j \in N_i^+ } d_j^+ ( d_j^+ + m_j^+ ) } \right) : v_i \in V(G) \bigg\}$. Moreover, the equality holds if and only if $G$ is a regular digraph or a bipartite semiregular digraph. (4) $q(G) \le \text{max} \big\{ \frac{1}{2} \left( d_i^+ + 2d_j^+ - 1 + \sqrt{ ( d_i^+ - 2d_j^+ + 1 )^2 + 4d_i^+ } \right) : ( v_j, v_i ) \in E(G) \big\}$. If the equality holds, then $G$ is a regular digraph or $G \in \Omega$, where $\Omega$ is a class of digraphs defined in this paper