Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

The Randić index and signless Laplacian spectral radius of graphs

Given a connected graph G, the Randić index R(G) is the sum of 1 √ d(u)d(v) over all edges {u, v} of G, where d(u) and d(v) are the degree of vertices u and v respectively. Let q(G) be the largest eigenvalue of the singless Laplacian matrix of G and n = |V (G)|. Hansen and Lucas (2010) made the following conjecture:

On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

On the Signless Laplacian Spectral Radius of Cacti

A cactus is a connected graph in which any two cycles have at most one vertex in common. We determine the unique graphs with maximum signless Laplacian spectral radius in the class of cacti with given number of cycles (cut edges, respectively) as well as in the class of cacti with perfect matchings and given number of cycles.

Sharp bounds for the signless Laplacian spectral radius of digraphs

On graphs whose spectral radius

The structure of graphs whose largest eigenvalue is bounded by 3 2 √ 2 (≈ 2.1312) is investigated. In particular, such a graph can have at most one circuit, and has a natural quipu structure.