Abstract:In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue −1, and determine all {C 3 , C 4 }-free connected graphs with three distinct distance eigenvalues of which the smallest one is equal to −3, which partially answers a problem posed by Koolen, Hayat and Iqbal [Linear Algebra Appl. 505 (2016) 97-108]. Furthermore, we characterize all trees with three distinct distance eigenvalues.
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