Graphs Of Rank Research Articles

Zonal graphs of small cycle rank

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order n and size m is m − n + 1 . Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank.

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer , it generates all required graphs with n vertices in time per graph and space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.

Complex unit gain graphs of rank 2

Let T be the group of all complex numbers z with |z|=1. A complex unit gain graph, or simply a T-gain graph, is a triple Φ=(G,T,φ) consisting of a graph G=(V,E), the circle group T and a gain function φ:E→→T such that φ(vivj)=φ(vjvi)−1=φ(vjvi)‾ for any adjacent vertices vi and vj. The adjacency matrix A(Φ) of the T-gain graph Φ=(G,T,φ) of order n is an n×n complex matrix (aij), where aij=aji‾=φ(vivj) if vi is adjacent to vj and aij=0 if otherwise. The rank of a T-gain graph Φ, denoted by r(Φ), is the rank of the adjacency matrix of Φ. Yu et al. [24] obtained some properties of inertia indexes of a T-gain graph and they characterized the T-gain unicyclic graphs with small positive or negative index. Lu [8] characterized the T-gain bicyclic graphs with rank 2,3 or 4. Motivated by above, we determine the rank of a T-gain unicyclic graph and classify the T-gain graphs with rank 2.

A program for finding all KMS states on the Toeplitz algebra of a higher-rank graph

The Toeplitz algebra of a finite graph of rank k carries a natural action of the torus Tk, and composing with an embedding of R in Tk gives a dynamics on the Toeplitz algebra. In this paper we describe a program for determining the simplex of KMS states associated to this dynamics at all inverse temperatures. For inverse temperatures larger than a critical value, the KMS states are well-understood, and this analysis is the first step in our program. At the critical inverse temperature, much less is known, and the second step in our program is an analysis of the KMS states at the critical value. This is the main technical contribution of the present paper. The third step in our program shows that the problem of finding the KMS states at inverse temperatures less than the critical value is equivalent to our original problem for a smaller graph. We test our program on a wide range of examples, including a very general family of graphs with three strongly connected components.

On the characterization of digraphs with given rank

The rank of a digraph is defined as the rank of its adjacency matrix. In this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2k. In particular, since the bipartite graphs of rank 2,4 and 6 are known, then we characterize digraphs of rank 1,2 and 3. The results are based on rank-preserving operations on digraphs such as splitting and fusion of vertices or twin-deletion and vertex-multiplication.

Chamber graphs of some geometries that are almost buildings

The global structure of the chamber graph of certain rank 3 geometries that are almost buildings is determined. Computer files containing extensive details of these graphs accompany this paper.

Characterization of oriented graphs of rank 2

Let D be an oriented graph, i.e., a digraph in which all arcs are not symmetric, with vertex set {1,…,n}. The adjacency matrix A=(aij) of D is the n×n matrix defined as aij=1 if ij is an arc of D and aij=0 if ij is not an arc of D. The rank of D is defined to be the rank of its adjacency matrix. In this paper, oriented graphs of rank 2 are characterized.

Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GP k )

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GP k , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GP k ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GP k ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GP k and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GP k is a graph of rank r and A(G)A(GP k ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

Kneser Ranks of Random Graphs and Minimum Difference Representations

Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v \mapsto A_v$ to the vertices $v\in V$ such that $A_u$ and $A_v$...

On the Local Structure of Mathon Distance-Regular Graphs

We study the structure of local subgraphs of distance-regular Mathon graphs of even valency. We describe some infinite series of locally Δ-graphs of this family, where Δ is a strongly regular graph that is the union of affine polar graphs of type “–,” a pseudogeometric graph for p G l (s, l), or a graph of rank 3 realizable by means of the van Lint–Schrijver scheme. We show that some Mathon graphs are characterizable by their intersection arrays in the class of vertex-transitive graphs.

Kneser ranks of random graphs and minimum difference representations

Every graph G=(V,E) is an induced subgraph of some Kneser graph of rank k, i.e., there is an assignment of (distinct) k-sets v↦Av to the vertices v∈V such that Au and Av are disjoint if and only if uv∈E. The smallest such k is called the Kneser rank of G and denoted by fKneser(G). As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant 0<p<1 there exist constants ci=ci(p)>0, i=1,2 such that with high probabilityc1n/(log⁡n)<fKneser(G)<c2n/(log⁡n). We apply this to other graph representations defined by Boros, Gurvich and Meshulam.A k-min-difference representation of a graph G is an assignment of a set Ai to each vertex i∈V(G) such that ij∈E(G)⇔min⁡{|Ai\\Aj|,|Aj\\Ai|}≥k. The smallest k such that there exists a k-min-difference representation of G is denoted by fmin(G). Balogh and Prince proved in 2009 that for every k there is a graph G with fmin(G)≥k. We prove that there are constants c1″,c2″>0 such that c1″n/(log⁡n)<fmin(G)<c2″n/(log⁡n) holds for almost all bipartite graphs G on n+n vertices.

Rank Reduction of Oriented Graphs by Vertex and Edge Deletions

In this paper we continue our study of graph modification problems defined by reducing the rank of the adjacency matrix of the given graph, and extend our results from undirected graphs to modifying the rank of skew-adjacency matrix of oriented graphs. An instance of a graph modification problem takes as input a graph G and a positive integer k, and the objective is to either delete k vertices/edges or edit k edges so that the resulting graph belongs to a particular family $$\mathcal{F}$$ of graphs. Given a fixed positive integer r, we define $$\mathcal{F}_r$$ as the family of oriented graphs where for each $$G\in \mathcal{F}_r$$ , the rank of the skew-adjacency matrix of G is at most r. Using the family $$\mathcal{F}_r$$ we do algorithmic study, both in classical and parameterized complexity, of the following graph modification problems: $$r$$ -Rank Vertex Deletion, $$r$$ -Rank Edge Deletion. We first show that both the problems are NP-Complete. Then we show that these problems are fixed parameter tractable (FPT) by designing an algorithm with running time $$2^{\mathcal{O}(k \log r)}n^{\mathcal{O}(1)}$$ for $$r$$ -Rank Vertex Deletion, and an algorithm for $$r$$ -Rank Edge Deletion running in time $$2^{\mathcal{O}(f(r) \sqrt{k} \log k )}n^{\mathcal{O}(1)}$$ . In addition to our FPT results we design polynomial kernels for these problems. Our main structural result, which is the fulcrum of all our algorithmic results, is that for a fixed integer r the size of any “reduced graph” in $$\mathcal{F}_r$$ is upper bounded by $$3^r$$ . This result is of independent interest and generalizes a similar result of Kotlov and Lovasz regarding reduced oriented graphs of rank r.

The BNS-invariant for some Artin groups of arbitrary circuit rank

Abstract We classify the Bieri–Neumann–Strebel invariant Σ 1 ⁢ ( G ) {\Sigma^{1}(G)} for a class of Artin groups with minimal graphs of arbitrary circuit rank.

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out (F_n) and Aut (F_n) from classes for groups \Gamma_{k,s} , where the \Gamma_{k,s} generalize Out (F_k)=\Gamma_{k,0} and Aut (F_k)=\Gamma_{k,1} . The symmetric group \mathfrak G_s acts on H_*(\Gamma_{k,s}) by permuting leaves, and for trivial rational coefficients we compute the \mathfrak G_s -module structure on H_*(\Gamma_{k,s}) completely for k \leq 2 . Assembling these classes then produces all the known nontrivial rational homology classes for Aut (F_n) and Out (F_n) with the possible exception of classes for n=7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut (F_n) and Out (F_n) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

Characterization of sub-long graphs of arbitrary rank

The rank of a graph is defined to be the rank of its adjacency matrix. A connected graph G of rank r is said to be sub-long if it has the second largest diameter, that is , among all connected graphs of rank r. In this paper, we characterize the sub-long graphs of rank r, for an arbitrary positive integer .

Automorphisms of a commuting graph of rank one upper triangular matrices

Let $F$ be a finite field, $n\geqslant 2$ an arbitrary integer, $\mathcal{M}_n(F)$ the set of all $n\times n$ matrices over $F$, and $\mathcal{U}_n^1(F)$ the set of all rank one upper triangular matrices of order $n$. For $\mathcal{S}\subseteq\mathcal{M}_n(F)$, denote $C(\mathcal{S})=\{X\in \mathcal{S} |\ XA=AX \ \hbox{for all}\ A\in \mathcal{S}\}$. The commuting graph of $\mathcal{S}$, denoted by $\Gamma(\mathcal{S})$, is the simple undirected graph with vertex set $\mathcal{S}\setminus C(\mathcal{S})$ in which for every two distinct vertices $A$ and $B$, $A\sim B$ is an edge if and only if $AB=BA$. In this paper, it is shown that any graph automorphism of $\Gamma(\mathcal{U}_n^1(F))$ with $n\geqslant 3$ can be decomposed into the product of an extremal automorphism, an inner automorphism, a field automorphism and a local scalar multiplication.

Hermitian adjacency spectrum and switching equivalence of mixed graphs

It is shown that an undirected graph G is cospectral with the Hermitian adjacency matrix of a mixed graph D obtained from a subgraph H of G by orienting some of its edges if and only if H=G and D is obtained from G by a four-way switching operation; if G is connected, this happens if and only if λ1(G)=λ1(D). All mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their Hermitian adjacency matrix. Several families of mixed graphs are found that are determined by their Hermitian spectrum in the sense that they are cospectral precisely to those mixed graphs that are switching equivalent to them.

Convexity in partial cubes: The hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves earlier results in the literature.On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations.Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane.To obtain the above results, we investigate convexity in partial cubes and obtain a new characterization of these graphs in terms of their lattice of convex subgraphs. This refines a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about tope graphs of rank 3 oriented matroids.

On order and rank of graphs

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most m(r)=2(r+2)/2?2 if r is even and m(r)=5·2(r?3)/2?2 if r is odd. In this article, we prove that if the conjecture is not true, then there would be a counterexample of rank at most 46. We also show that every reduced graph of rank r has at most 8m(r)+14 vertices.