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Suppose that the vertex set of a connected graph $G$ is$V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distancesbetween $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$diagonal matrix with its $(i,i)$-entry equal to $Tr_{G}(v_{i})$ and $D(G)$ bethe distance matrix of $G$. Then $Q_{D}(G)=Tr(G)+D(G)$ is the distance signlessLaplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distancesignless Laplacian spectral radius of $G$. In this paper we give some grafttransformations on distance signless Laplacian spectral radius of the graphsand use them to characterize the graphs with the minimum and maximal distancesignless Laplacian spectral radius among non-starlike and non-caterpillartrees.

The effect of a graft transformation on distance signless Laplacian spectral radius of the graphs