Inverse nodal problems on quantum tree graphs

We consider inverse nodal problems for the Sturm–Liouville operators on the tree graphs. Can only dense nodes distinguish the tree graphs? In this paper it is shown that the data of dense-nodes uniquely determines the potential (up to a constant) on the tree graphs. This provides interesting results for an open question implied in the paper.

Fractal behaviours of networks induced on infinite tree structures by random walks

Tree graphs such as Cayley trees provide a stage to support the self-organization of fractal networks by the flow of walkers from the root vertex to the outermost shell of the tree graph. This network model is a typical example that demonstrates the ability of a random process on a network to generate fractality. However, the finite scale of the tree structure assumed in the model restricts the size of fractal networks. In this study, we removed the restriction on the size of the trees by introducing a lifetime τ (number of steps of random walks) of walkers. As a result, we successfully induced a size-independent fractal structure on a tree graph without a boundary. Our numerical results show that the mean number of offspring d b of the original tree structure determines the value of the fractal box dimension db through the relation d b — 1 = (n b — 1) -θ . The lifetime τ controls the presence or absence of small-world and scale-free properties. The ideal fractal behaviour can be maintained by selecting an appropriate value of τ. The numerical results contribute to the development of a systematic method for generating fractal small-world and scale-free networks while controlling the value of the fractal box dimension. Unlike other models that use recursive rules to generate self-similar structures, this model specifically produces small-world fractal networks with scale-free properties.

The use of B-splines to represent the topography of river networks

This work presents a new extension to B-Splines that enables them to model functions on directed tree graphs such as non-braided river networks. The main challenge of the application of B-splines to graphs is their definition in the neighbourhood of nodes with more than two incident edges. Achieving that the B-splines are continuous at these points is non-trivial. For both, simplification reasons and in view of our application, we limit the graphs to directed tree graphs. To fulfil the requirement of continuity, the knots defining the B-Splines need to be located symmetrically along the edges with the same direction. With such defined B-Splines, we approximate the topography of the Mekong River system from scattered height data along the river. To this end, we first test and validate successfully the method with synthetic water level data, with and without added annual signal. The quality of the resulting heights is assessed besides others by means of root mean square errors (RMSE) and mean absolute differences (MAD). The RMSE values are 0.26 m and 1.05 m without and with added annual variation respectively and the MAD values are even lower with 0.11 m and 0.60 m. For the second test, we use real water level observations measured by satellite altimetry. Again, we successfully estimate the river topography, but also discuss the short comings and problems with unevenly distributed data. The unevenly distributed data leads to some very large outliers close to the upstream ends of the rivers tributaries and in regions with rapidly changing topography such as the Mekong Falls. Without the outlier removal the standard deviation of the resulting heights can be as large as 50 m with a mean value of 15.73 m. After the outlier removal the mean standard deviation drops to 8.34 m.

Smoothed Nested Testing on Directed Acyclic Graphs

Summary We consider the problem of multiple hypothesis testing when there is a logical nested structure to the hypotheses. When one hypothesis is nested inside another, the outer hypothesis must be false if the inner hypothesis is false. We model the nested structure as a directed acyclic graph, including chain and tree graphs as special cases. Each node in the graph is a hypothesis and rejecting a node requires also rejecting all of its ancestors. We propose a general framework for adjusting node-level test statistics using the known logical constraints. Within this framework, we study a smoothing procedure that combines each node with all of its descendants to form a more powerful statistic. We prove a broad class of smoothing strategies can be used with existing selection procedures to control the familywise error rate, false discovery exceedance rate, or false discovery rate, so long as the original test statistics are independent under the null. When the null statistics are not independent but are derived from positively-correlated normal observations, we prove control for all three error rates when the smoothing method is arithmetic averaging of the observations. Simulations and an application to a real biology dataset demonstrate that smoothing leads to substantial power gains.

ON EVEN-TO-ODD MEAN LABELING OF SOME TREES

Let G=(V(G), E(G)) be a connected graph with order |V(G)|=p and size |E(G)|=q. A graph G is said to be even-to-odd mean graph if there exists a bijection function phi:V(G) to {2, 4, ..., 2p} such that the induced mapping phi^*:E(G) to {3, 5, ..., 2p-1} defined by phi^*(uv)=[phi(u)+phi(v)]/2 for all uv element of E(G) is also bijective. The function is called an even-to-odd mean labeling of graph . This paper aimed to introduce a new technique in graph labeling. Hence, the concepts of even-to-odd mean labeling has been evaluated for some trees. In addition, we examined some properties of tree graphs that admits even-to-odd mean labeling and discussed some important results.

Sharp lower bounds on the sum-connectivity index of quasi-tree graphs

The sum-connectivity index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. A graph [Formula: see text] is called quasi-tree, if there exists [Formula: see text] such that [Formula: see text] is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs.