Geodesic
Reflexive line graphs of trees
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 2, pp. 447-464.

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A graph is reflexive if the second largest eigenvalue of its adjacency matrix is less than or equal to 2. In this paper, we characterize trees whose line graphs are reflexive. It turns out that these trees can be of arbitrary order -- they can have either a unique vertex of arbitrary degree or pendant paths of arbitrary lengths, or both. Since the reflexive line graphs are Salem graphs, we also relate some of our results to the Salem (graph) numbers.
Classification : 05C76, 05C05, 05C50
Keywords: line graph, subdivision graph, adjacency matrix, second largest eigenvalue, reflexive graph, Salem graph 
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     author = {Simi\'c, Slobodan K. and \v{Z}ivkovi\'c, Dejan and An{\dj}eli\'c, Milica and da Fonseca, Carlos M.},
     title = {Reflexive line graphs of trees},
     journal = {Journal of Algebraic Combinatorics},
     pages = {447--464},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2016},
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Simić, Slobodan K.; Živković, Dejan; Anđelić, Milica; da Fonseca, Carlos M. Reflexive line graphs of trees. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 2, pp. 447-464. http://geodesic.mathdoc.fr/item/JAC_2016__43_2_a1/