The Maximum Second Zagreb Spectral Radius of Trees, Unicyclic Graphs and Bicyclic Graphs

The second Zagreb matrix of a graph $G$ is $Z(G)=D(G)A(G)D(G)$, where $D(G)$ and $A(G)$ are the degree matrix and the adjacency matrix of $G$. For many degree-based weighted adjacency matrices, the trees, unicyclic graphs and bicyclic graphs with the largest spectral radius have a vertex adjacent to all others, and Mei et al. conjectured this for every such matrix on unicyclic graphs. Jiang et al. showed that $Z(G)$ is a counterexample. We prove that, for $q\in\{0,1,2\}$ and every $n\ge4$, the connected graph of order $n$ and cyclomatic number $q$ maximizing the spectral radius of $Z(G)$ is unique, and is obtained from two adjacent vertices with $q$ common neighbours by attaching $n-q-2$ pendent vertices to them as evenly as possible. The only exceptions are the star for $q=0$ and $n\le7$, and the triangle with one pendent vertex at each vertex for $q=1$ and $n=6$. This confirms a conjecture of Jiang et al.\ and all three parts of a conjecture of Shen and Shan for this matrix, with best possible thresholds on $n$. For $n\ge8$, the maximum value is $\lfloor (n+q)^2/4\rfloor+(n+q)/2+7q-2+O(1/n)$, whereas the star gives only $(n-1)^{3/2}$. A key tool is a determinant criterion that reduces spectral comparisons to sign conditions on explicit polynomials.