1,045 research outputs found
On the imaginary parts of chromatic root
While much attention has been directed to the maximum modulus and maximum
real part of chromatic roots of graphs of order (that is, with
vertices), relatively little is known about the maximum imaginary part of such
graphs. We prove that the maximum imaginary part can grow linearly in the order
of the graph. We also show that for any fixed , almost every
random graph in the Erd\"os-R\'enyi model has a non-real root.Comment: 4 figure
On the Roots of Chromatic Polynomials
AbstractIt is proved that the chromatic polynomial of a connected graph with n vertices and m edges has a root with modulus at least (m−1)/(n−2); this bound is best possible for trees and 2-trees (only). It is also proved that the chromatic polynomial of a graph with few triangles that is not a forest has a nonreal root and that there is a graph with n vertices whose chromatic polynomial has a root with imaginary part greater thann/4
On the Real Roots of Domination Polynomials
A dominating set of a graph of order is a subset of the vertices
of such that every vertex is either in or adjacent to a vertex of .
The domination polynomial is defined by where is
the number of dominating sets in with cardinality . In this paper we
show that the closure of the real roots of domination polynomials is
On the Split Reliability of Graphs
A common model of robustness of a graph against random failures has all
vertices operational, but the edges independently operational with probability
. One can ask for the probability that all vertices can communicate ({\em
all-terminal reliability}) or that two specific vertices (or {\em terminals})
can communicate with each other ({\em two-terminal reliability}). A relatively
new measure is {\em split reliability}, where for two fixed vertices and
, we consider the probability that every vertex communicates with one of
or , but not both. In this paper, we explore the existence for fixed numbers
and of an {\em optimal} connected -graph
for split reliability, that is, a connected graph with vertices
and edges for which for any other such graph , the split reliability of
is at least as large as that of , for {\em all} values of . Unlike the similar problems for all-terminal and two-terminal
reliability, where only partial results are known, we completely solve the
issue for split reliability, where we show that there is an optimal
-graph for split reliability if and only if , , or
.Comment: 12 pages, 9 figure
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