TY: JOUR
T1 - Some new considerations about double nested graphs
A1 - Andelic, M.
A1 - Andrade, E.
A1 - Cardoso, D. M.
A1 - Fonseca, C. M. da
A1 - Simic, S. K.
A1 - Tosic, D. V.
N2 - In the set of all connected graphs with fixed order and size, the graphs with
maximal index are nested split graphs, also called threshold graphs. It was recently (and
independently) observed in [F.K.Bell, D. Cvetkovi´c, P. Rowlinson, S.K. Simi´c, Graphs
for which the largest eigenvalue is minimal, II, Linear Algebra Appl. 429 (2008)] and
[A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs,
Electron. J. Combin. 15 (2008), #144] that double nested graphs, also called bipartite
chain graphs, play the same role within class of bipartite graphs. In this paper we study
some structural and spectral features of double nested graphs. In studying the spectrum
of double nested graphs we rather consider some weighted nonnegative matrices (of
significantly less order) which preserve all positive eigenvalues of former ones. Moreover,
their inverse matrices appear to be tridiagonal. Using this fact we provide several new
bounds on the index (largest eigenvalue) of double nested graphs, and also deduce some
bounds on eigenvector components for the index. We conclude the paper by examining
the questions related to main versus non-main eigenvalues.
UR - https://ria.ua.pt/handle/10773/15062
Y1 - 2015
PB - Elsevier