389 research outputs found
An improvement of sufficient condition for -leaf-connected graphs
For integer a graph is called -leaf-connected if and given any subset with always has a
spanning tree such that is precisely the set of leaves of Thus a
graph is -leaf-connected if and only if it is Hamilton-connected. In this
paper, we present a best possible condition based upon the size to guarantee a
graph to be -leaf-connected, which not only improves the results of Gurgel
and Wakabayashi [On -leaf-connected graphs, J. Combin. Theory Ser. B 41
(1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for
-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also
extends the result of Xu, Zhai and Wang [An improvement of spectral conditions
for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key
approach is showing that an -closed non--leaf-connected graph must
contain a large clique if its size is large enough. As applications, sufficient
conditions for a graph to be -leaf-connected in terms of the (signless
Laplacian) spectral radius of or its complement are also presented.Comment: 15 pages, 2 figure
Spectral radius and spanning trees of graphs
For integer a spanning -ended-tree is a spanning tree with at
most leaves. Motivated by the closure theorem of Broersma and Tuinstra
[Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237],
we provide tight spectral conditions to guarantee the existence of a spanning
-ended-tree in a connected graph of order with extremal graphs being
characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with
constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we
also present tight spectral conditions for the existence of a spanning tree
with leaf degree at most in a connected graph of order with extremal
graphs being determined, where is an integer
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
Expanders Are Universal for the Class of All Spanning Trees
Given a class of graphs F, we say that a graph G is universal for F, or
F-universal, if every H in F is contained in G as a subgraph. The construction
of sparse universal graphs for various families F has received a considerable
amount of attention. One is particularly interested in tight F-universal
graphs, i.e., graphs whose number of vertices is equal to the largest number of
vertices in a graph from F. Arguably, the most studied case is that when F is
some class of trees.
Given integers n and \Delta, we denote by T(n,\Delta) the class of all
n-vertex trees with maximum degree at most \Delta. In this work, we show that
every n-vertex graph satisfying certain natural expansion properties is
T(n,\Delta)-universal or, in other words, contains every spanning tree of
maximum degree at most \Delta. Our methods also apply to the case when \Delta
is some function of n. The result has a few very interesting implications. Most
importantly, we obtain that the random graph G(n,p) is asymptotically almost
surely (a.a.s.) universal for the class of all bounded degree spanning (i.e.,
n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a
constant. Moreover, a corresponding result holds for the random regular graph
of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq
n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and
the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are
a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence
of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta,
we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs
with clique number at most five. Finally, we show robustness of random graphs
with respect to being universal for T(n,\Delta) in the context of the
Maker-Breaker tree-universality game.Comment: 25 page
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