Edge-Complement Graphs – Another Approach

In this paper, we investigate the problem of clustering XML documents based on their structure. We represent the paths in an XML document as a multiset and use the symmetric difference operation on multisets to define certain metrics. These metrics are then used to obtain a measure of similarity between any two documents in a collection. Our technique was successfully applied to real and synthesized XML documents yielding high-quality clusterings.

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Graceful labeling of power graph of group Z2k−1 × Z4

Asian-European Journal of Mathematics ◽

10.1142/s1793557122501212 ◽

2021 ◽

pp. 2250121

Author(s):

Amit Sehgal ◽

Neeraj Takshak ◽

Pradeep Maan ◽

Archana Malik

Keyword(s):

Finite Group ◽

Abelian Group ◽

Finite Abelian Group ◽

Simple Graph ◽

Graceful Labeling ◽

Power Graph ◽

Vertex Set ◽

A Finite Group

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

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Local dissymmetry on graphs and related algebraic structures

International Journal of Algebra and Computation ◽

10.1142/s0218196719500607 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1499-1526 ◽

Cited By ~ 3

Author(s):

G. Chiaselotti ◽

T. Gentile ◽

F. Infusino

Keyword(s):

Algebraic Structure ◽

Binary Operation ◽

Simple Graph ◽

Symmetric Difference ◽

Graph Structure ◽

Algebraic Structures ◽

Vertex Set ◽

Structure Formula

We use the set symmetric difference between vertex subsets of a finite undirected simple graph [Formula: see text] to define a binary operation ∘ on the vertex set of a new graph [Formula: see text], that contains [Formula: see text] as subgraph and whose vertices are non-empty vertex subsets of [Formula: see text]. We show how the binary operation ∘ determines an algebraic structure on [Formula: see text] that is strictly related to the graph structure of [Formula: see text]. In fact, we show that [Formula: see text] agrees with [Formula: see text] and, next, we provide several characterizations for the algebraic structure [Formula: see text] when the graph [Formula: see text] is connected and locally dissymmetric.

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Hamiltonicity of Token Graphs of Some Join Graphs

Symmetry ◽

10.3390/sym13061076 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1076

Author(s):

Luis Enrique Adame ◽

Luis Manuel Rivera ◽

Ana Laura Trujillo-Negrete

Keyword(s):

Infinite Family ◽

Simple Graph ◽

Symmetric Difference ◽

Hamiltonian Graphs ◽

Vertex Set

Let G be a simple graph of order n with vertex set V(G) and edge set E(G), and let k be an integer such that 1≤k≤n−1. The k-token graph G{k} of G is the graph whose vertices are the k-subsets of V(G), where two vertices A and B are adjacent in G{k} whenever their symmetric difference A▵B, defined as (A∖B)∪(B∖A), is a pair {a,b} of adjacent vertices in G. In this paper we study the Hamiltonicity of the k-token graphs of some join graphs. We provide an infinite family of graphs, containing Hamiltonian and non-Hamiltonian graphs, for which their k-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which it is proven the Hamiltonicity of their k-token graphs, for any 2<k<n−2.

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Common Neighborhood Energy of Commuting Graphs of Finite Groups

Symmetry ◽

10.3390/sym13091651 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1651

Author(s):

Rajat Kanti Nath ◽

Walaa Nabil Taha Fasfous ◽

Kinkar Chandra Das ◽

Yilun Shang

Keyword(s):

Abelian Group ◽

Finite Groups ◽

Regular Polygon ◽

Undirected Graph ◽

Abelian Groups ◽

Simple Graph ◽

Central Quotient ◽

Vertex Set ◽

The Common ◽

Group Of Symmetries

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

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