Vertex-edge Incidence Research Articles

ONTRANSFORMEDGRAPHS

The network systems and graphical analysis through the study of structural characteristics is a vast field of growing importance in research. Topological indices have asignificant and crucial role in the study of structure–property relationships. In this paper, we examine constructional transformed networks constructed by unique vertex-edge incidence and mutual adjacency associations. Expressions for the first and second hyper Zagreb indices and co-indices of these transformed networks and their complements are obtained.

The sum of hamming distance for some composite graphs relative to vertex-edge incidence matrix

Given C be a vertex-edge incidence matrix of a graph Z. The Hamming distance between two rows C(x, :) and C(y, :) of C is the number of coloumn ℓ such that C (x, ℓ) ≠ C (y, ℓ). In this paper, we discuss the sum of Hamming distances between all pairs of rows of C. We present a formula for the sum of the number of edges and the number of vertices of the graph Z. We then use this formula to determine the sum of Hamming distance for composite graphs such as union of graphs, joint of graphs, corona product of graphs and cartesian product of graphs

Rank of the Vertex-Edge Incidence Matrix of r-Out Hypergraphs

We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, \cite{ACO}, \cite{CFP}. For fields of higher order, the co-rank is typically Poisson distributed.

On the Chromatic Index of the Signed Generalized Petersen Graph GP(n,2)

Let G be a graph and σ:E(G)→{+1,−1} be a mapping. The pair (G,σ), denoted by Gσ, is called a signed graph. A (proper) l-edge coloring γ of Gσ is a mapping from each vertex–edge incidence of Gσ to Mq such that γ(v,e)=−σ(e)γ(w,e) for each edge e=vw, and no two vertex–edge incidences have the same color; that is, γ(v,e)≠γ(v,f). The chromatic index is the minimal number q such that Gσ has a proper q-edge coloring, denoted by χ′(Gσ). In 2020, Behr proved that the chromatic index of a signed graph is its maximum degree or maximum plus one. In this paper, we considered the chromatic index of the signed generalized Petersen graph GP(n,2) and show that its chromatic index is its maximum degree for most cases. In detail, we proved that (1) χ′(GPσ(n,2))=3 if n≡3 mod 6(n≥9); (2) χ′(GPσ(n,2))=3 if n=2p(p≥4).

The inverse of the incidence matrix of a unicyclic graph

The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse of the incidence matrix of an even unicyclic graph is presented solving an open problem.

Querying a Matrix through Matrix-Vector Products

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

Balanced flows for transshipment problems

A transshipment problem (G,d,λ) is modeled by a directed graph G=(V,E) with weighted vertices (nodes) d=(dv∣v∈V) and weighted directed edges (arcs) λ=(λe∣e∈E) interpreted as follows: G is a communication or transportation network, for example, a pipeline; each arc e∈E is a one-way communication line, road or pipe of capacity λe, while every vertex v∈V is a node of production dv>0, consumption dv<0, or just transition dv=0. A non-negative flow x=(xe∣e∈E) is called weakly feasible if for each v∈V the algebraic sum of flows, over all directed edges incident to v, equals dv, or shorter, if AGx=d, where AG is the vertex-edge incidence matrix of G. A weakly feasible flow x is called feasible if xe≤λe for all e∈E. We consider weakly feasible but not necessarily feasible flows, that is, inequalities xe>λe are allowed. However, such an excess is viewed as unwanted (dangerous) and so we minimize the excess ratio vector r=(re=xe/λe∣e∈E) lexicographically. More precisely, first, we look for weakly feasible flows minimizing the maximum of re over all e∈E; among all such flows we look for those that minimize the second largest coordinate of r, etc. Clearly, in |E| such steps we obtain a unique lexmin vector r. We show that the corresponding balanced flow is also unique and represents the lexmin solution for problem (G,d,λ). Using the Gale–Hoffman inequalities, we construct it in polynomial time, provided vectors d and λ are rational.For symmetric digraphs the problem was solved by Gurvich and Gvishiani in 1984. Here we extend this result to arbitrary directed graphs. Furthermore, we simplify the algorithm and proofs applying the classic criterion of existence of a feasible flow for (G,d,λ) obtained by Gale and Hoffman in late 1950s.

Estimating cellular redundancy in networks of genetic expression

Networks of genetic expression can be modeled by hypergraphs with the additional structure that real coefficients are given to each vertex-edge incidence. The spectra, i.e. the multiset of the eigenvalues, of such hypergraphs, are known to encode structural information of the data. We show how these spectra can be used, in particular, in order to give an estimation of cellular redundancy, a novel measure of gene expression heterogeneity, of the network. We analyze some simulated and real data sets of gene expression for illustrating the new method proposed here.

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

ABSTRACT A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs.

On the Rank of a Random Binary Matrix

We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \choose k}$ such columns.&#x0D; We obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn/k$. The matrix $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m&lt;m^*$ the rank of $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$.&#x0D; In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$, the expected length of a minimum weight spanning tree tends to $\zeta(3)\sim 1.202$.

Spectra of cycle and path families of oriented hypergraphs

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of either +1 or −1. This generalizes signed graphs to a hypergraph setting and simultaneously provides a natural definition for a signed adjacency which is used to define the adjacency and Laplacian matrices. Many properties of these matrices are known, but there are no nontrivial families of oriented hypergraphs with their spectrum determined. In this paper we define and study hypergraph families that are analogous to cycles and paths in graphs and signed graphs. The adjacency and Laplacian eigenvalues for some of these families is determined.

Moore–Penrose inverse of incidence matrix of graphs with complete and cyclic blocks

Let Γ be a graph with n vertices, where each edge is given an orientation and let Q be the vertex–edge incidence matrix of Γ. Suppose that Γ has a cut-vertex v and Γ−v=Γ[V1]∪Γ[V2]. We obtain a relation between the Moore–Penrose inverse of the incidence matrix of Γ and of the incidence matrices of the induced subgraphs Γ[V1∪{v}] and Γ[V2∪{v}]. The result is used to give a combinatorial interpretation of the Moore–Penrose inverse of the incidence matrix of a graph whose blocks are either cliques or cycles. Moreover we obtain a description of minors of the Moore–Penrose inverse of the incidence matrix when the rows are indexed by cut-edges. The results generalize corresponding results for trees in the literature.

Diametral Paths in Total Graphs of Paths, Cycles and Stars

The diametral path of a graph is the shortest path between two vertices which has length equal to diameter of that graph. Total graph of a graph is a graph that has vertices representing all vertices and edges of the original graph and edges representing every vertex-vertex adjacency, edge-edge adjacency and edge-vertex incidence. In this paper, the number of diametral paths is determined for the paths, cycles and stars and their total graphs.

Moore–Penrose inverse of the incidence matrix of a distance regular graph

Let Γ be a graph with n vertices, where each edge is given an orientation and let Q be the vertex-edge incidence matrix. We obtain a formula for the Moore–Penrose inverse of Q, when Γ is a distance regular graph. The formula is illustrated by examples.

Graph theory for analyzing pair-wise data: application to geophysical model parameters estimated from interferometric synthetic aperture radar data at Okmok volcano, Alaska

Graph theory is useful for analyzing time-dependent model parameters estimated from interferometric synthetic aperture radar (InSAR) data in the temporal domain. Plotting acquisition dates (epochs) as vertices and pair-wise interferometric combinations as edges defines an incidence graph. The edge-vertex incidence matrix and the normalized edge Laplacian matrix are factors in the covariance matrix for the pair-wise data. Using empirical measures of residual scatter in the pair-wise observations, we estimate the relative variance at each epoch by inverting the covariance of the pair-wise data. We evaluate the rank deficiency of the corresponding least-squares problem via the edge-vertex incidence matrix. We implement our method in a MATLAB software package called GraphTreeTA available on GitHub (https://github.com/feigl/gipht). We apply temporal adjustment to the data set described in Lu et al. (Geophys Res Solid Earth 110, 2005) at Okmok volcano, Alaska, which erupted most recently in 1997 and 2008. The data set contains 44 differential volumetric changes and uncertainties estimated from interferograms between 1997 and 2004. Estimates show that approximately half of the magma volume lost during the 1997 eruption was recovered by the summer of 2003. Between June 2002 and September 2003, the estimated rate of volumetric increase is (6.2 , pm , 0.6) times 10^6~mathrm{m}^3/mathrm{year} . Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 eruption. Although we demonstrate the approach using volumetric rates of change, our formulation in terms of incidence graphs applies to any quantity derived from pair-wise differences, such as range change, range gradient, or atmospheric delay.

A characterization of oriented hypergraphic balance via signed weak walks

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −1, and each adjacency is signed the negative of the product of the incidences. An oriented hypergraph is balanced if the product of the adjacencies in each circle is positive.We provide a combinatorial interpretation for entries of kth power of the oriented hypergraphic Laplacian via the number of signed weak walks of length k. Using closed weak walks we prove a new characterization of balance for oriented hypergraphs and matrices that generalizes Harary's Theorem for signed graphs.

Extremal Edge Polytopes

The "edge polytope" of a finite graph $G$ is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For $k =2, 3, 5$ we determine the maximal number of vertices of $k$-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.

Spectral properties of oriented hypergraphs

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the signless Laplacian of a graph and also suggests a natural way to define the adjacency and Laplacian matrices of a hypergraph. Some results presented generalize both graph and signed graph results to a hypergraphic setting.

An oriented hypergraphic approach to algebraic graph theory

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or -1. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.

Nowhere-harmonic colorings of graphs

Proper vertex colorings of a graph are related to its boundary map ∂ 1 \partial _1 , also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, L = ∂ 1 ∂ 1 t L=\partial _1 \partial _1^t , a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss several examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings.