Maxima of the Q-index of graphs with pairs of forbidden subgraphs and minimum degree δ≥2

Let Q(G)=D(G)+A(G) denote the signless Laplacian matrix (or Q(G)-matrix) of the graph G. Denote by q(G) (or Q(G)-index) the signless Laplacian spectral radius of the graph G. Let θ(l1,l2,l3) denote the theta graph which consists of two vertices connected by three internally disjoint paths with length l1, l2 and l3. For odd n≥5, Fn denotes the graph consisting of n-12 triangles which intersect in exactly one common vertex. For even n≥6, Fn denotes the graph obtained by hanging an edge to the maximal degree vertex of Fn-1. In this paper, we firstly show that if G is a {C3,C4}-free graph with order n≥5 and minimum degree δ≥2, then q(G)≤n+32, unless G≅C5. Secondly, we show that if G is a {θ(1,2,2),F5}-free graph with order n≥6 and minimum degree δ≥2, then q(G)≤n, unless G≅G3 for n=6 or G≅Kt,n-t for n≥6 and 2≤t≤n-2. Finally, we show that if G is a {θ(1,2,2),θ(1,2,3)}-free graph with size m≥9 and minimum degree δ≥2, then q(G)≤q(F2m+33) for m=3k,k≥3, unless G≅F2m+33.