Some results on the majorization theorem of connected graphs

Let pi = (d (1), d (2), ..., d (n) ) and pi' = (d' (1), d' (2), ..., d' (n) ) be two non-increasing degree sequences. We say pi is majorizated by pi', denoted by pi aS(2) pi', if and only if pi not equal pi', I pound (i=1) (n) d (i) = I pound (i=1) (n) d' (i) , and I pound (i=1) (j) d (i) a parts per thousand currency sign I pound (i=1) (j) d' (i) for all j = 1, 2, ..., n. Weuse C (pi) to denote the class of connected graphs with degree sequence pi. Let rho(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553-557 (2009)] and [BA +/- yA +/- koglu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if pi and pi' are two different non-increasing degree sequences of unicyclic graphs with pi aS(2) pi', G and G' are the unicyclic graphs with the greatest spectral radii in C (pi) and C' (pi) , respectively, then rho(G) < rho(G').