Geodesic
Walk counts and the spectral radius of graphs
Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 40 (2015) no. 1 
Dragan Stevanović. Walk counts and the spectral radius of graphs. Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 40 (2015) no. 1. http://geodesic.mathdoc.fr/item/BASS_2015_40_1_a3/
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     author = {Dragan Stevanovi\'c},
     title = {Walk counts and the spectral radius of graphs},
     journal = {Bulletin de l'Acad\'emie serbe des sciences. Classe des sciences math\'ematiques et naturelles},
     pages = {33 - 57},
     year = {2015},
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     number = {1},
     url = {http://geodesic.mathdoc.fr/item/BASS_2015_40_1_a3/}
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Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

We develop a new method that uses walk counts for comparing spectral radii of graphs similar in a precisely defined fashion. The method is applied to the cases where a path-like or a star-like structure is coalesced to a graph, in order to prove weak inequality in the conjectured inequality of Belardo, Li Marzi and Simić, and to resolve the Brualdi-Solheid problem for the classes of graphs consisting of rooted products with the same rooted graph.