Several Families Of Graphs Research Articles

New Lower Bound for the Minimum Sum Coloring Problem

The Minimum Sum Coloring Problem (MSCP) is an NP-Hard problem derived from the graph coloring problem (GCP) and has practical applications in different domains such as VLSI design, distributed resource allocation, and scheduling. There exist few exact solutions for MSCP, probably due to its search space much more elusive than that of GCP. On the contrary, much effort is spent in the literature to develop upper and lower bounds for MSCP. In this paper, we borrow a notion called motif, that was used in a recent work for upper bounding the minimum number of colors in an optimal solution of MSCP, to develop a new algebraic lower bound called for MSCP. Experiments on standard benchmarks for MSCP and GCP show that this new lower bound is substantially better than the existing lower bounds for several families of graphs.

On vertex and edge H-irregularity strengths of graphs

We introduce two new graph characteristics, the edge [Formula: see text]-irregularity strength and the vertex [Formula: see text]-irregularity strength of a graph. We estimate the bounds of these parameters and determine their exact values for several families of graphs namely, paths, ladders and fans.

The maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph

In this paper we introduce a parameter Mm(G), defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph G. We develop basic properties of Mm(G) and compute it for several families of graphs.

Group distance magic and antimagic graphs

Given a graph $G$ with $n$ vertices and an Abelian group $A$ of order $n$, an $A$-distance antimagic labelling of $G$ is a bijection from $V(G)$ to $A$ such that the vertices of $G$ have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of $A$) of the labels assigned to its neighbours. An {$A$-distance magic labelling} of $G$ is a bijection from $V(G)$ to $A$ such that the weights of all vertices of $G$ are equal to the same element of $A$. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.

χ_D(G), |Aut(G)|, and a variant of the motion lemma

The Distinguishing Chromatic Number of a graph G , denoted χ D ( G ) , was first defined in K. L. Collins and A. N. Trenk, The distinguishing chromatic number, Electron. J. Combin. 13 (2006), #R16, as the minimum number of colors needed to properly color G such that no non-trivial automorphism ϕ of the graph G fixes each color class of G . In this paper, We prove a lemma that may be considered a variant of the Motion lemma of A. Russell and R. Sundaram, A note on the asympotics and computational complexity of graph distinguishability, Electron. J. Combin. 5 (1998), #R23, and use this to give examples of several families of graphs which satisfy χ D ( G ) = χ ( G ) + 1 . We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also describe families of graphs with (relatively) very small automorphism groups which satisfy χ D ( G ) = χ ( G ) + 1 , for arbitrarily large values of χ ( G ) . We describe non-trivial families of bipartite graphs that satisfy χ D ( G ) > r for any positive integer r .

Prime vertex labelings of several families of graphs

A simple and connected $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$, such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs including cycle pendant stars, cycle chains, prisms, and generalized books.

A note on Distance Magic and Distance antimagic graphs

Let G=(V,E) be a graph of order n. The graph G is said to be distance magic if there exists a bijection f:V(G)→{1,2,…,n} such that for all v∈V, w(v)=∑u∈N(v)f(u) is a constant, called vertex magic constant. The graph G is said to be distance anti-magic if w(u)≠w(v) for all u, v in G. We prove that several families of graphs are distance antimagic.

The failed zero forcing number of a graph

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the color change rule results in all vertices joining S. The color change rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u joins S in the next iteration. In this paper, we introduce a new graph parameter, the failed zero forcing number of a graph. The failed zero forcing number of G, F(G), is the maximum cardinality of any set of vertices on which repeated applications of the color change rule will never result in all vertices joining the set. We establish bounds on the failed zero forcing number of a graph, both in general and for connected graphs. We also classify connected graphs that achieve the upper bound, graphs whose failed zero forcing numbers are zero or one, and unusual graphs with smaller failed zero forcing number than zero forcing number. We determine formulas for the failed zero forcing numbers of several families of graphs and provide a lower bound on the failed zero forcing number of the Cartesian product of two graphs. We conclude by presenting open questions about the failed zero forcing number and zero forcing in general.

On edge irregularity strength of graphs

For a simple graph G, a vertex labeling ϕ:V(G)→{1,2,…,k} is called k-labeling. The weight of an edge xy in G, denoted by wϕ(xy), is the sum of the labels of end vertices x and y, i.e. wϕ(xy)=ϕ(x)+ϕ(y). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f there is wϕ(e)≠wϕ(f). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G).In this paper, we estimate the bounds of the edge irregularity strength and determine the exact value for several families of graphs.

Graphs that allow all the eigenvalue multiplicities to be even

Let G be an undirected graph on n vertices and let S(G) be the set of all n×n real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The Inverse Eigenvalue Problem for a graph G is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in S(G). This question is, in general, hard to answer and several variations were studied, most notably the minimum rank problem. In this paper we introduce the problem of determining for which graphs G there exists a matrix in S(G) whose characteristic polynomial is a square, i.e. the multiplicities of all its eigenvalues are even. We solve this question for several families of graphs.

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

AbstractIn an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .

Classes of graphs with minimum skew rank 4

The minimum skew rank of a simple graph G is the smallest possible rank among all real skew-symmetric matrices whose (i,j)-entry is nonzero if and only if the edge joining i and j is in G. It is known that a graph has minimum skew rank 2 if and only if it consists of a complete multipartite graph and some isolated vertices. Some necessary conditions for a graph to have minimum skew rank 4 are established, and several families of graphs with minimum skew rank 4 are given. Linear algebraic techniques are developed to show that complements of trees and certain outerplanar graphs have minimum skew rank 4.

New Spectral Bounds on the Chromatic Number Encompassing all Eigenvalues of the Adjacency Matrix

The purpose of this article is to improve existing lower bounds on the chromatic number $\chi$. Let $\mu_1,\ldots,\mu_n$ be the eigenvalues of the adjacency matrix sorted in non-increasing order.First, we prove the lower bound $\chi \ge 1 + \max_m\{\sum_{i=1}^m \mu_i / -\sum_{i=1}^m \mu_{n - i +1}\}$ for $m=1,\ldots,n-1$. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the Hoffman lower bound.Second, we conjecture the lower bound $\chi \ge 1 + s^+ / s^-$, where $s^+$ and $s^-$ are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the bound $\chi \ge s^+/s^-$. We show that the conjectured lower bound is true for several families of graphs. We also performed various searches for a counter-example, but none was found.Our proofs rely on a new technique of considering a family of conjugates of the adjacency matrix, which add to the zero matrix, and use majorization of spectra of self-adjoint matrices.We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected.All these bounds are also valid when we replace the adjacency matrix $A$ by $W * A$ where $W$ is an arbitrary self-adjoint matrix and $*$ denotes the Schur product, that is, entrywise product of $W$ and $A$.

Complexity Analysis of Quantum Walk Based Search Algorithms

We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O($\sqrt{n}$ log($n$)) elementary 2-qubit quantum gate operations to find a marked node.

Product irregularity strength of certain graphs

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E ( G ) → {1, 2, …, m } is called product - irregular , if all product degrees p d G ( v ) = ∏ e ∋ v w ( e ) are distinct. The goal is to obtain a product - irregular labeling that minimizes the maximal label. This minimal value is called the product irregularity strength and denoted p s ( G ) . We give the exact values of p s ( G ) for several families of graphs, as complete bipartite graphs K m , n , where 2 ≤ m ≤ n ≤ ( m + 2) choose 2, some families of forests, including complete d -ary trees, and other graphs with δ ( G ) = 1 .

Distance edge coloring and collision-free communication in wireless sensor networks

Motivated by the problem of link scheduling in wireless sensor networks where different sensors have different transmission and interference ranges and may be mobile, we study the problem of “distance edge coloring” of graphs, which is a generalization of proper edge coloring. Let Gbe a graph modeling a sensor network. An l-distance edge coloring of Gis a coloring of the edges of Gsuch that any two edges within distance lof each other are assigned different colors. The parameter lis chosen, so that the links corresponding to two edges that are assigned the same color do not interfere. We investigate the l-distance edge coloring problem on several families of graphs that can be used as topologies in sensor deployment. We focus on determining the minimum number of colors needed and optimal coloring algorithms. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

The Number of Ways to Assemble a Graph

Motivated by the question of how macromolecules assemble,the notion of an assembly tree of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs. In some natural special cases, we use a powerful recent result of Zeilberger and Apagodu to provide recurrence relations for the diagonal of the relevant multivariate generating functions, and we use a result of Wimp and Zeilberger to find very precise asymptotic formulae for the coefficients of these diagonals. Future directions for reseach, as well as open questions, are suggested.

Stack Domination Density

There are infinite sequences of graphs {G n } where |G n | = n such that the minimal dominating sets for G i × H fall into predictable patterns, in light of which ? (G n × H) may be nearly linear in n; the coefficient of linearity may be regarded as the average density of the dominating set in the H-fibers of the product. The specific cases where the sequence {G n } consists of cycles or path is explored in detail, and the domination density of the Grotzsch graph is calculated. For several other sequences {G n }, the limit of this density can be seen to exist; in other cases the ratio $${\frac{\gamma (G_n \times H)}{\gamma (G_n)}}$$ proves to be of greater interest, and also exists for several families of graphs.

Computing global offensive alliances in Cartesian product graphs

A global offensive alliance in a graph G is a set S of vertices with the property that every vertex not belonging to S has at least one more neighbor in S than it has outside of S. The global offensive alliance number of G, γo(G), is the minimum cardinality of a global offensive alliance in G. A set S of vertices of a graph G is a dominating set for G if every vertex not belonging to S has at least one neighbor in S. The domination number of G, γ(G), is the minimum cardinality of a dominating set of G. In this work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we also prove that γo(G□H)≥γ(G)γo(H)2 for any graphs G and H and we show that if G has an efficient dominating set, then γo(G□H)≥γ(G)γo(H). Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.

On the metric dimension of infinite graphs

A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.