Non-isomorphic caterpillars with identical subtree data

Eisenstat, David; Gordon, Gary

doi:10.1016/j.disc.2006.01.022

Cited by 16 publications

(12 citation statements)

References 7 publications

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“…In , pairs of trees are constructed with the property that they have the same number of subtrees of size k for all k . (In fact, the pairs of trees constructed there have the same number of subtrees with k edges and l leaves, for all k and l .)…”

Section: Preliminariesmentioning

confidence: 99%

Subtrees of graphs

Chin

Gordon

MacPhee

et al. 2018

Journal of Graph Theory

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We study a new graph invariant, the sequence {sk} of the number of k‐edge subtrees of a graph. We compute the mean subtree size for several classes of graphs, concentrating on complete graphs, complete bipartite graphs, and theta graphs, in particular. We prove that the ratio of spanning trees to all subtrees in Kn approaches (1/e)(1/e)=0.692201⋯, and give a related formula for Kn,n. We also connect the number of subtrees of Kn that contain a given subtree to the hyperbinomial transform. For theta graphs, we find formulas for the mean subtree size (approximately 23n) and the mode (approximately 22n) of the unimodal sequence {sk}. The main tool is a subtree generating function.

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Section: Preliminariesmentioning

confidence: 99%

Subtrees of graphs

Chin

Gordon

MacPhee

et al. 2018

Journal of Graph Theory

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“…Indeed, the two trees T 1 , T 2 shown in Fig. 2 share the same subtree polynomial; this is a special case of a theorem of Eisenstat and Gordon [4]. On the other hand, X T 1 = X T 2 .…”

Section: Conjecture 7 (Z-integrality) Let μ N Be a Partition Then mentioning

confidence: 92%

On distinguishing trees by their chromatic symmetric functions

Martin

Morin

Wagner

2008

Journal of Combinatorial Theory, Series A

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Let T be an unrooted tree. The chromatic symmetric function X T , introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of T . The subtree polynomial S T , first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of T by their numbers of edges and leaves. We prove that S T = Φ, X T , where ·,· is the Hall inner product on symmetric functions and Φ is a certain symmetric function that does not depend on T . Thus the chromatic symmetric function is a stronger isomorphism invariant than the subtree polynomial. As a corollary, the path and degree sequences of a tree can be obtained from its chromatic symmetric function. As another application, we exhibit two infinite families of trees (spiders and some caterpillars), and one family of unicyclic graphs (squids) whose members are determined completely by their chromatic symmetric functions.

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“…Also, they proved that this polynomial uniquely determines rooted trees. For unrooted trees however, it is shown in [15] that certain classes of caterpillars have the same polynomials assigned to them. However, interestingly, we prove that in each poset, in many cases the trees have unique polynomials.…”

Section: Tutte-like Polynomials For Gaussian Trees In Posetsmentioning

confidence: 99%

Topological and Algebraic Properties for Classifying Unrooted Gaussian Trees under Privacy Constraints

Moharrer¹,

Wei²,

Amariucai³

et al. 2014

2015 IEEE Global Communications Conference (GLOBECOM)

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In this paper, our objective is to find out how topological and algebraic properties of unrooted Gaussian tree models determine their security robustness, which is measured by our proposed max-min information (MaMI) metric. Such metric quantifies the amount of common randomness extractable through public discussion between two legitimate nodes under an eavesdropper attack. We show some general topological properties that the desired max-min solutions shall satisfy. Under such properties, we develop conditions under which comparable trees are put together to form partially ordered sets (posets). Each poset contains the most favorable structure as the poset leader, and the least favorable structure. Then, we compute the Tutte-like polynomial for each tree in a poset in order to assign a polynomial to any tree in a poset. Moreover, we propose a novel method, based on restricted integer partitions, to effectively enumerate all poset leaders. The results not only help us understand the security strength of different Gaussian trees, which is critical when we evaluate the information leakage issues for various jointly Gaussian distributed measurements in networks, but also provide us both an algebraic and a topological perspective in grasping some fundamental properties of such models.

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Non-isomorphic caterpillars with identical subtree data

Cited by 16 publications

References 7 publications

Subtrees of graphs

Subtrees of graphs

On distinguishing trees by their chromatic symmetric functions

Topological and Algebraic Properties for Classifying Unrooted Gaussian Trees under Privacy Constraints

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