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Keywords

Maximum degree, Optimizing, Spectral radius, Tree

Abstract

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let \$T^*(n,\Delta ,l)\$ be the tree which minimizes the spectral radius of all trees of order \$n\$ with exactly \$l\$ vertices of maximum degree \$\Delta \$. In this paper, \$T^*(n,\Delta ,l)\$ is determined for \$\Delta =3\$, and for \$l\le 3\$ and \$n\$ large enough. It is proven that for sufficiently large \$n\$, \$T^*(n,3,l)\$ is a caterpillar with (almost) uniformly distributed legs, \$T^*(n,\Delta ,2)\$ is a dumbbell, and \$T^*(n,\Delta ,3)\$ is a tree consisting of three distinct stars of order \$\Delta \$ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

vol31_pp335-361.pdf (288 kB)
Abstract

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