Graphs with diameter [formula omitted] minimizing the spectral radius

Abstract

The spectral radius ρ(G) of a graph G is the largest eigenvalue of its adjacency matrix A(G). For a fixed integer e⩾1, let Gn,n-emin be a graph with minimal spectral radius among all connected graphs on n vertices with diameter n-e. Let Pn1,n2,…,nt,pm1,m2,…,mt 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, Gn,n-emin were determined in the literature. Cioabaˇ et al. [2] conjectured for fixed e⩾6, Gn,n-emin is in the family Pn,e={P2,1,…,1,2,n-e+12,m2,…,me-4,n-e-2∣2<m2<⋯<me-4<n-e-2}. For e=6,7, they conjectured Gn,n-6min=P2,1,2,n-52,⌈D-12⌉,D-2 and Gn,n-7min=P2,1,1,2,n-62,⌊D+23⌋,D-⌊D+23⌋,D-2. 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 T(k1,k2,…,ke-4)=P2,1,…,1,2,n-e+12,m2,…,me-4,n-e-2 with ki=mi+1-mi-1, for 1⩽i⩽e-4. (Here we assume m1=2 and me-3=n-e-2.)Let s=∑i=1e-4ki+2e-4. For any integer e⩾6 and sufficiently large n, we proved that Gn,n-emin must be one of the trees T(k1,k2,…,ke-4) with the parameters satisfying ⌊s⌋-1⩽kj⩽⌊s⌋⩽ki⩽⌈S⌉+1 for j=1,e-4 and i=2,…,e-5. Moreover, 0⩽ki-kj⩽2 for 2⩽i⩽e-5,j=1,e-4; and |ki-kj|⩽1 for 2⩽i,j⩽e-5. These results are best possible as shown by cases e=6,7,8, where Gn,n-emin are completely determined here. Moreover, if n-6 is divisible by e-4 and n is sufficiently large, then Gn,emin=T(k-1,k,k,…,k,k,k-1) where k=n-6e-4-2.