458 research outputs found
Graphs with small diameter determined by their -spectra
Let be a connected graph with vertex set
. The distance matrix is the matrix indexed by the vertices of where denotes the
distance between the vertices and . Suppose that
are the distance
spectrum of . The graph is said to be determined by its -spectrum if
with respect to the distance matrix , any graph having the same spectrum
as is isomorphic to . In this paper, we give the distance characteristic
polynomial of some graphs with small diameter, and also prove that these graphs
are determined by their -spectra
Maxima of the -index of non-bipartite -free graphs
A classic result in extremal graph theory, known as Mantel's theorem, states
that every non-bipartite graph of order with size contains a triangle. Lin, Ning and Wu [Comb. Probab.
Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for
given order Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a
spectral version for fixed size In this paper, we prove -spectral
versions of Mantel's theorem.Comment: 14 pages, 4 figure
Spectral radius, fractional -factor and ID-factor-critical graphs
Let be a graph and be a function. For any two
positive integers and with , a fractional -factor of
with the indicator function is a spanning subgraph with vertex set
and edge set such that for
any vertex , where and E_{G}(v)=\{e\in
E(G)| e~\mbox{is incident with}~v~\mbox{in}~G\}. A graph is
ID-factor-critical if for every independent set of whose size has the
same parity as , has a perfect matching. In this paper, we
present a tight sufficient condition based on the spectral radius for a graph
to contain a fractional -factor, which extends the result of Wei and
Zhang [Discrete Math. 346 (2023) 113269]. Furthermore, we also prove a tight
sufficient condition in terms of the spectral radius for a graph with minimum
degree to be ID-factor-critical.Comment: 14 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
Laplacian eigenvalue distribution, diameter and domination number of trees
For a graph with domination number , Hedetniemi, Jacobs and
Trevisan [European Journal of Combinatorics 53 (2016) 66-71] proved that
, where means the number of Laplacian
eigenvalues of in the interval . Let be a tree with diameter
. In this paper, we show that . However, such a
lower bound is false for general graphs. All trees achieving the lower bound
are completely characterized. Moreover, for a tree , we establish a relation
between the Laplacian eigenvalues, the diameter and the domination number by
showing that the domination number of is equal to if and only if
it has exactly Laplacian eigenvalues less than one. As an
application, it also provides a new type of trees, which show the sharpness of
an inequality due to Hedetniemi, Jacobs and Trevisan
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