1,276,058 research outputs found
On the diameter and incidence energy of iterated total graphs
The total graph of , is the graph whose set of vertices is
the union of the sets of vertices and edges of , where two vertices are
adjacent if and only if they stand for either incident or adjacent elements in
. Let , the total graph of . For
, the iterated total graph of , , is
defined recursively as If
is a connected graph its diameter is the maximum distance between any pair
of vertices in . The incidence energy of is the sum of the
singular values of the incidence matrix of . In this paper for a given
integer we establish a necessary and sufficient condition under which
. In addition, bounds for the
incidence energy of the iterated graph are obtained,
provided to be a regular graph. Finally, new families of non-isomorphic
cospectral graphs are exhibited
The t-stability number of a random graph
Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at most t.
The t-stability number of G is the maximum order of a t-stable set in G. We
investigate the typical values that this parameter takes on a random graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed
non-negative integer t, we show that, with probability tending to 1 as n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic expression for
the expected number of t-stable sets of order k. We derive this expression by
performing a precise count of the number of graphs on k vertices that have
maximum degree at most k. Using the above results, we also obtain asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the vertex set of
the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version
apart from formatting and a minor amendment to Lemma 8 (and its proof on p.
21
Circumference and Pathwidth of Highly Connected Graphs
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t
has treewidth at most t-1. Under the additional assumption of 2-connectivity,
such graphs have bounded pathwidth, which is a qualitatively stronger result.
Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007]
who showed that every graph without k disjoint cycles of length at least t has
bounded treewidth (as a function of k and t). Our main result states that,
under the additional assumption of (k + 1)- connectivity, such graphs have
bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover,
examples show that (k + 1)-connectivity is required for bounded pathwidth to
hold. These results suggest the following general question: for which values of
k and graphs H does every k-connected H-minor-free graph have bounded
pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure
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