Graphs with diameter n-e minimizing the spectral radius

The spectral radius ρ(G) of a graph G is the largest eigenvalue of its adjacency matrix A(G). For a fixed integer , let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter n-e. Let be a tree obtained from a path of p vertices (0∼1∼2∼⋯∼(p-1)) by linking one pendant path Pni at mi for each i∈{1,2,…,t}. For e=1,2,3,4,5, were determined in the literature. Cioabaˇ et al. [2] conjectured for fixed , is in the family . For e=6,7, they conjectured and . In this paper, we settle their conjectures positively. Note that any tree in Pn,e is uniquely determined by its internal path lengths. For any e-4 non-negative integers k1,k2,…,ke-4, let with ki=mi+1-mi-1, for . (Here we assume m1=2 and me-3=n-e-2.)Let . For any integer e⩾6 and sufficiently large n, we proved that must be one of the trees T(k1,k2,…,ke-4) with the parameters satisfying for j=1,e-4 and i=2,…,e-5. Moreover, for and for . These results are best possible as shown by cases e=6,7,8, where are completely determined here. Moreover, if n-6 is divisible by e-4 and n is sufficiently large, then where .