Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs

Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs $\mathcal{F}$, a graph is called $\mathcal{F}$-free if it contains no member of $\mathcal{F}$ as a subgraph. The theta graph $θ(l_1,\ldots,l_k)$ consists of $k$ internally disjoint paths of lengths $l_1,\ldots,l_k$ with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all $\{C_3,C_4\}$-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all $\{θ(1,2,2),θ(1,2,3)\}$-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.