Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The signless Laplacian matrix of a graph G is defined as Q ( G ) = D ( G ) + A ( G ), where D ( G ) is the degree-diagonal matrix of G and A ( G ) is the adjacency matrix of G . The multiplicity of an eigenvalue mu of Q ( G ) is denoted by m Q ( G , mu ). Let G = C ( T 1 , ... , T g ) be a unicyclic graph with a perfect matching, where Ti is a rooted tree attached at the vertex v i of the cycle C g . It is proved that Q ( G ) has 2 as an eigenvalue if and only if g + t is divisible by 4, where t is the number of T i of even orders. Another main result of this article gives a characterization for a connected graph G with a perfect matching such that m Q ( G , 2) = 0 ( G ) + 1, where 0 ( G ) = | E ( G ) | - | V ( G ) | + 1 is the cyclomatic number of G . (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.