Block-cutvertex trees and block-cutvertex partitions

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme–Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

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“…The tree obtained in Lemma 6 is similar to the "block-cutvertex tree" defined in [HP66] (see [Bar02]). Proof.…”

Section: A2mentioning

confidence: 93%

Patterns in Khovanov link and chromatic graph homology

Sazdanovic

Scofield

2018

J. Knot Theory Ramifications

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“…BCtree) was proposed, it has been drawing researchers' attention and concern worldwide, not only in the field of mathematics and computer science, but also in chemistry and brain science. See Barefoot [5], Mkrtchyan [17], Misiołek and Chen [18], Paton [19], Gagarin and Labelle [20], Nakayama and Fujiwara [21], Yang and Wang [22] It is well known that the Wiener index among trees on n vertices is minimized by the star K 1,n−1 and is maximized by the n-vertex path P n , see Entringer et al [23], or Lovász [24].…”

Section: Wiener Odd (Even) Index On Bc-trees a Wiener Odd (Evenmentioning

confidence: 99%

“…BC-tree have many special properties and applications in graph theory, chemistry, computer and brain science, see [3], [4], [5], [6], [7].…”

Section: Introductionmentioning

confidence: 99%

Wiener odd and even indices on BC-Trees

Liang

Liu

et al. 2013

2013 Third World Congress on Information and Communication Technologies (WICT 2013)

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Abstract-Corresponding to the concepts of Wiener index and distance of the vertex, in this paper, we present the concepts of Wiener odd (even) index of G as sum of the distances between all pairs of vertices of G satisfying the distances are all odd (even) and denote them W odd (G) and W odd (G) respectively. Based on the concepts of the two indices, we prove theoretically that Wiener odd index is not more than its even index for general BC-Trees. Closed formulae of the two indices are also provided for path BC-tree, star, k-extending star tree and caterpillar BC-tree. Meanwhile, the extreme values of W odd (T ) of n vertices BC-trees are characterized as well.

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“…They may contain, for instance, some cut-vertices of the graph that are not in the set C. However if C is the set of all the cut-vertices of the graph (that is blocks are exactly the 2-connected components), T is then related to the block-cutvertex graph introduced by Gallai, Harary and Prins in the '60 (cf. [Plu68,Bar02]). The main result of this section is the following theorem:…”

Section: Distance Labeling Schemes Combinationmentioning

confidence: 99%

Distance labeling scheme and split decomposition

Gavoille

Paul

2003

Discrete Mathematics

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A distance labeling scheme is a distributed data-structure designed to answer queries about distance between any two vertices of a graph G. The data-structure consists in a label L(x, G) assigned to each vertex x of G such that the distance d G (x, y) between any two vertices x and y can be estimated as a function f (L(x, G), L(y, G)). In this paper, by the use of split decomposition of graphs, we combine several types of distance labeling schemes. This yields to optimal label length schemes for the family of distance-hereditary graphs and for other families of graphs, allowing distance estimation in constant time once the labels have been constructed.

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Block-cutvertex trees and block-cutvertex partitions

Cited by 17 publications

References 1 publication

Patterns in Khovanov link and chromatic graph homology

Patterns in Khovanov link and chromatic graph homology

Wiener odd and even indices on BC-Trees

Distance labeling scheme and split decomposition

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