32 research outputs found
Maxima Q-index of graphs with forbidden odd cycles
Let be the -index (the largest eigenvalue of the
signless Laplacian) of . Let be the graph obtained by joining each
vertex of a complete graph of order to each vertex of an independent set of
order The main result of this paper is the following theorem:
Let and be a graph of order . If has no
then unless
This result proves the odd case of the conjecture in [M.A.A. de Freitas, V.
Nikiforov, and L. Patuzzi, Maxima of the -index: forbidden -cycle and
-cycle, \emph{Electron. J. Linear Algebra }26 (2013), 905-916.]Comment: The new version contains a new proof of Lemma 10, as the previous one
had a gap, which was pointed out by Dr. Bo Nin
Maxima of the Q-index: forbidden 4-cycle and 5-cycle
This paper gives tight upper bounds on the largest eigenvalue q(G) of the
signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let
F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1}
with an edge hanged to its center. It is shown that if G is a graph of order n,
with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of
a complete graph of order k and an independent set of order n-k. It is shown
that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless
G=S_{n,k}. It is shown that these results are significant in spectral extremal
graph problems. Two conjectures are formulated for the maximum q(G) of graphs
with forbidden cycles.Comment: 12 page
An asymptotically tight bound on the Q-index of graphs with forbidden cycles
Let G be a graph of order n and let q(G) be that largest eigenvalue of the
signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2,
then G contains cycles of length l whenever 2<l<2k+3. This bound is
asymptotically tight. It implies an asymptotic solution to a recent conjecture
about the maximum q(G) of a graph G with no cycle of a specified length.Comment: 10 pages. Version 2 takes care of some mistakes in version
Maxima of the Q-index: graphs without long paths
This paper gives tight upper bound on the largest eigenvalue q(G) of the
signless Laplacian of graphs with no paths of given order. The main ingredient
of our proof is a stability result of its own interest, about graphs with large
minimum degree and with no long paths. This result extends previous work of Ali
and Staton.Comment: 10 page
Number of walks and degree powers in a graph
This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its
vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees
Maxima of the Q-index: forbidden even cycles
Let be a graph of order and let be the largest
eigenvalue of the signless Laplacian of . Let be the graph
obtained by joining each vertex of a complete graph of order to each vertex
of an independent set of order and let be the graph
obtained by adding an edge to
It is shown that if and is a graph of order
with no cycle of length then unless This result completes the proof
of a conjecture of de Freitas, Nikiforov and Patuzzi.Comment: 16 page