The Minimal Number of Subtrees with a Given Degree Sequence

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

A note on extremal trees with degree conditions

We investigate to what extent the degree sequence of a directed network constrains the number of driver nodes. We develop a pair of algorithms that take a directed degree sequence as input and aim to output a network with the maximum or minimum number of driver nodes. We find an upper bound for the maximum and a lower bound for the minimum, and show that the algorithms achieve these bounds for all real and model networks, with few exceptions characterized by tiny system size and heterogeneous degree distributions. Applying these algorithms to a broad range of real networks, we show the gap between the upper and lower bounds can vary dramatically across different degree sequences. Thus, we introduce the notion of \bl{structural control} complexity to capture how much more difficult it is to control a specific network beyond what is required given its degree sequence, suggesting additional structure is present. Using model networks, we numerically and analytically investigate how typical features of the degree distribution affect the range of required driver nodes. We find that the minimum is determined by the number of sources or sinks, while the maximum is strongly affected by the presence of hubs.

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree.

On the extremal eccentric connectivity index of graphs

Controlling complex network is an essential problem in network science and engineering. Recent advances indicate that the controllability of complex network is dependent on the network's topology. Liu and Barabási, et.al speculated that the degree distribution was one of the most important factors affecting controllability for arbitrary complex directed network with random link weights. In this paper, we analysed the effect of degree distribution to the controllability for the deterministic networks with unweighted and undirected. We introduce a class of deterministic networks with identical degree sequence, called (x,y)-flower. We analysed controllability of the two deterministic networks ((1, 3)-flower and (2, 2)-flower) by exact controllability theory in detail and give accurate results of the minimum number of driver nodes for the two networks. In simulation, we compare the controllability of (x,y)-flower networks. Our results show that the family of (x,y)-flower networks have the same degree sequence, but their controllability is totally different. So the degree distribution itself is not sufficient to characterize the controllability of deterministic networks with unweighted and undirected.

We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number are analyzed in stochastic DAGs in both the cases of a regular and power law degree sequence. For each such sequence, we give asymptotic bounds (and in certain instances, precise values) for the green number.

On subtrees of trees

A generalized ‐independent set is a set of vertices such that the induced subgraph contains no trees with ‐vertices, and the generalized ‐independence number is the cardinality of a maximum ‐independent set in . Zito proved that the maximum number of maximum generalized 2‐independent sets in a tree of order is if is odd, and if is even. Tu et al. showed that the maximum number of maximum generalized 3‐independent sets in a tree of order is if , and if , and if and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized ‐independent sets in a tree for a general integer . As applications, we show that the maximum number of generalized 4‐independent sets in a tree of order is and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4‐independent sets.

Digital microfluidic systems (DMFS) are a class of lab-on-a-chip systems that manipulate individual droplets of chemicals on an array of electrodes. The biochemical analyses are performed by repeatedly moving, mixing, and splitting droplets on the electrodes. In this paper, we characterize the tree structure of biochemical analyses and identify their minimum resource requirements, towards the design of cost and space-efficient biochips. Mixers and storage units are two primary functional resources on a DMFS biochip; mixers mix and split droplets while storage units store droplets on the chip for subsequent processing. Additional DMFS resources include input and output units and transportation paths. We present an algorithm to compute, for a given number of mixers M, the minimum number of storage units f(M) for an input analysis using its tree structure, and design a corresponding scheduling algorithm to perform the analysis. We characterize the variation of the M-depth of a tree (i.e., its minimum number of storage units f(M)) with M, and use it to calculate the minimum total size (the number of electrodes) of mixers and storage units. We prove that the smallest chip for an arbitrary analysis uses one mixer and f(1) storage units. Finally, we demonstrate our results on two example biochemical analyses and design the smallest chip for a biochemical analysis with a complete tree structure of depth 4. These are the first results on the least resource requirements of DMFS for biochemical analyses, and can be used for the design and selection of chips for arbitrary biochemical analyses.KeywordsTime SlotBiochemical AnalysisSchedule AlgorithmResource RequirementOutput UnitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Switching fabrics may be (1) multistage architectures with shuffle interstage interconnections (Omega, Banyan) or the interconnection of nearest-neighbour switches (Spanke-Benes network, crossbar) or (2) tree structures where the latter presently are considered to be the most suitable from the point of view of a low attentuation of signals, crosstalk and insertion loss.

GLOUDS: Representing tree-like graphs

A multiple-interval representation of a simple graph G assigns each vertex a union of disjoint real intervals so that vertices are adjacent if and only if their assigned sets intersect. The total interval number$I(G)$ is the minimum of the total number of intervals used in such a representation of G. For triangle-free graphs, $I(G) = | E(G) |+t(G)$, where $t(G)$ is the minimum number of pairwise edge-disjoint trails that together contain an endpoint of each edge. This yields the NP-completeness of testing $I(G) = | E(G) | + 1$ (even for triangle-free 3-regular planar graphs) and an alternative proof that HAMILTONIAN CYCLE is NP-complete for line graphs. It also yields a linear-time algorithm to compute $I(G)$ for trees and a characterization of the trees requiring $| E(G) | + t$ intervals for fixed t. Further corollaries include the Aigner-Andreae bound of $I(G) \leq \lfloor {(5n - 3)/4} \rfloor $ for n-vertex trees (achieved by subdividing every edge of a star), a characterization of the extremal trees, and a shorter proof of the extremal bound $\lfloor {(5m + 2)/4} \rfloor $ for connected graphs.

OF THE DISSERTATION Some Problems On Discrete Geometry and Combinatorics by Lei Wang Dissertation Directors: William Steiger and Mario Szegedy Let K be a convex body in the plane. It is known that K can never be partitioned into seven regions of equal area by three non-concurrent lines. We will be concerned with a partition of K by three non-concurrent lines such that the ratio of the area of smallest region to the area of biggest region is maximum. We call this an optimal balanced partition at K. We show that the best possible ratio is achieved when K is a triangle and we characterize the optimal balanced partition in this case. We conjecture that the condition holds for optimal balanced partitions of all convex bodies but can only prove a weaker result. In the second part of the thesis, we switch to the zigzag problem. We are given a set of n points in R2 and seek the minimum number of line segments required for a polygonal chain (or a simple polygonal chain) to traverse all the points. We show an n/2 + O(n/ log n) upper bound if self-intersection is allowed and an n − ⌊n−2 8 ⌋ upper bound if self-intersection is not allowed. The third part of this thesis is about finding the optimally balanced forward degree sequence of a graph. The final part studies the optimal solutions for some variants of the Towers of Hanoi problem.