scholarly journals Edit Distance and its Computation

10.37236/744 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
József Balogh ◽  
Ryan Martin
Keyword(s):  
Bipartite Graph ◽  
Edit Distance ◽  
Complete Bipartite Graph ◽  
New Method ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties ◽  
Szemeredi's Regularity Lemma

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemerédi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm Forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Turán's theorem, which may be of independent interest.

scholarly journals The Edit Distance Function and Symmetrization

10.37236/2262 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Ryan R. Martin
Keyword(s):  
Distance Function ◽  
Edit Distance ◽  
Symmetric Difference ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets.  The distance between a graph, G, and a hereditary property, ℋ, is the minimum of the distance between G and each G'∈ℋ.  The edit distance function of ℋ is a function of p∈[0,1] and is the limit of the maximum normalized distance between a graph of density p and ℋ.This paper utilizes a method due to Sidorenko [Combinatorica 13(1), pp. 109-120], called "symmetrization", for computing the edit distance function of various hereditary properties.  For any graph H, Forb(H) denotes the property of not having an induced copy of H.  This paper gives some results regarding estimation of the function for an arbitrary hereditary property. This paper also gives the edit distance function for Forb(H), where H is a cycle on 9 or fewer vertices.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

scholarly journals A Note on the Speed of Hereditary Graph Properties

10.37236/644 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Vadim V. Lozin ◽  
Colin Mayhill ◽  
Victor Zamaraev
Keyword(s):  
Practical Importance ◽  
Bipartite Graphs ◽  
Hereditary Property ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Graph Properties ◽  
Graph Property ◽  
Vertex Set ◽  
Split Graphs ◽  
Hereditary Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

Some Results on Annihilating-ideal Graphs

2016 ◽  
Vol 59 (3) ◽  
pp. 641-651
Author(s):  
Farzad Shaveisi
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Complete Bipartite Graph ◽  
Independence Number ◽  
Prime Ideals ◽  
Reduced Ring ◽  
Vertex Set ◽  
Minimal Prime

AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

scholarly journals Mallows permutations as stable matchings

2020 ◽  
pp. 1-25
Author(s):  
Omer Angel ◽  
Alexander E. Holroyd ◽  
Tom Hutchcroft ◽  
Avi Levy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Stable Matching ◽  
Countable Family ◽  
Stable Matchings ◽  
Uncountable Family ◽  
Edge Crossing ◽  
Vertex Set ◽  
Infinite Intervals ◽  
The Law

Abstract We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

scholarly journals The Smith and Critical Groups of the Square Rook's Graph and its Complement

10.37236/5442 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joshua E. Ducey ◽  
Jonathan Gerhard ◽  
Noah Watson
Keyword(s):  
Normal Form ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
Critical Group ◽  
Critical Groups ◽  
Complete Graphs ◽  
Vertex Set

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column.  This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$.  In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement.  This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs.  In doing so we verify a 1986 conjecture of Rushanan.

scholarly journals Codegree Threshold for Tiling Balanced Complete $3$-Partite $3$-Graphs and Generalized $4$-Cycles

10.37236/9061 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Xinmin Hou ◽  
Boyuan Liu ◽  
Yue Ma
Keyword(s):  
Bipartite Graph ◽  
Error Term ◽  
Complete Bipartite Graph ◽  
Linear Term ◽  
Asymptotic Values ◽  
F Factor ◽  
Vertex Set ◽  
Error Terms ◽  
Vertex Disjoint

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex-disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every  $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling.  Mycroft (JCTA, 2016) determined  the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$ and conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we determine the exact value of $t_2(n, K_{m,m}^{3})$, where $K_{m,m}^{3}$ (defined by Mubayi and Verstraëte, JCTA, 2004) is the 3-graph obtained from the complete bipartite graph $K_{m,m}$ by replacing each vertex in one part by a 2-elements set. Note that $K_{2,2}^{3}$ is  the well known  generalized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinct edges $A, B, C, D$ with $A\cup B= C\cup D$ and $A\cap B=C\cap D=\emptyset$). The result confirms Mycroft's conjecture for $K_{m,m}^{3}$. Moreover, we improve the error term $o(n)$ to a sub-linear term when $F=K^3(m)$ and show that the sub-linear term is tight for $K^3(2)$, where $K^3(m)$ is the complete $3$-partite $3$-graph with each part of size $m$.

The coloring of the cozero-divisor graph of a commutative ring

2020 ◽  
Vol 12 (03) ◽  
pp. 2050023
Author(s):  
S. Akbari ◽  
S. Khojasteh
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Local Ring ◽  
Chromatic Number ◽  
Upper Bound ◽  
Complete Bipartite Graph ◽  
Clique Number ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Non Local

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

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