4 research outputs found
The shape of node reliability
Given a graph whose edges are perfectly reliable and whose nodes each
operate independently with probability the node reliability of
is the probability that at least one node is operational and that the
operational nodes can all communicate in the subgraph that they induce. We
study analytic properties of the node reliability on the interval
including monotonicity, concavity, and fixed points. Our results show a stark
contrast between this model of network robustness and models that arise from
coherent set systems (including all-terminal, two-terminal and K-terminal
reliability).Comment: 21 page
The expected subtree number index in random polyphenylene and spiro chains
Subtree number index \emph{STN}(G) of a graph is the number of nonempty
subtrees of . It is a structural and counting based topological index that
has received more and more attention in recent years. In this paper we first
obtain exact formulas for the expected values of subtree number index of random
polyphenylene and spiro chains, which are molecular graphs of a class of
unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover,
we establish a relation between the expected values of the subtree number
indices of a random polyphenylene and its corresponding hexagonal squeeze. We
also present the average values for subtree number indices with respect to the
set of all polyphenylene and spiro chains with hexagons.Comment: 16pages, 3 figure
Wiener index, number of subtrees, and tree eccentric sequence
The eccentricity of a vertex in a connected graph is the distance
between and a vertex farthest from it; the eccentric sequence of is the
nondecreasing sequence of the eccentricities of . In this paper, we
determine the unique tree that minimises the Wiener index, i.e. the sum of
distances between all unordered vertex pairs, among all trees with a given
eccentric sequence. We show that the same tree maximises the number of subtrees
among all trees with a given eccentric sequence, thus providing another example
of negative correlation between the number of subtrees and the Wiener index of
trees. Furthermore, we provide formulas for the corresponding extreme values of
these two invariants in terms of the eccentric sequence. As a corollary to our
results, we determine the unique tree that minimises the edge Wiener index, the
vertex-edge Wiener index, the Schulz index (or degree distance), and the Gutman
index among all trees with a given eccentric sequence.Comment: 14 page
Subtrees and independent subsets in unicyclic graphs and unicyclic graphs with fixed segment sequence
In the study of topological indices two negative correlations are well known:
that between the number of subtrees and the Wiener index (sum of distances),
and that between the Merrifield-Simmons index (number of independent vertex
subsets) and the Hosoya index (number of independent edge subsets). That is,
among a certain class of graphs, the extremal graphs that maximize one index
usually minimize the other, and vice versa. In this paper, we first study the
numbers of subtrees in unicyclic graphs and unicyclic graphs with a given
girth, further confirming its opposite behavior to the Wiener index by
comparing with known results. We then consider the unicyclic graphs with a
given segment sequence and characterize the extremal structure with the maximum
number of subtrees. Furthermore, we show that these graphs are not extremal
with respect to the Wiener index. We also identify the extremal structures that
maximize the number of independent vertex subsets among unicyclic graphs with a
given segment sequence, and show that they are not extremal with respect to the
number of independent edge subsets. These results may be the first examples
where the above negative correlation failed in the extremal structures between
these two pairs of indices.Comment: 21 pages 6 figure