New skew equienergetic oriented graphs

Treelike Comparability Graphs

The mutual exclusion scheduling problem for permutation and comparability graphs

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.

Optimum skew energy of a tournament

Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are 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]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals 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]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

Let $G=(V, E)$ be a simple undirected graph. For a given set $L\subset \mathbb{R}$, a function $\omega: E \longrightarrow L$ is called an $L$-flow. Given a vector $\gamma \in \mathbb{R}^V$, $\omega$ is a $\gamma$-$L$-flow if for each $v\in V$, the sum of the values on the edges incident to $v$ is $\gamma(v)$. If $\gamma(v)=c$, for all $v\in V$, then the $\gamma$-$L$-flow is called a $c$-sum $L$-flow. In this paper, the existence of $\gamma$-$L$-flows for various choices of sets $L$ of real numbers is studied, with an emphasis on 1-sum flows. Let $L$ be a subset of real numbers containing $0$ and denote $L^*:=L\setminus \{0\}$. Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of $0$-sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a $1$-sum $\mathbb{R}^*$-flow or a $1$-sum $\mathbb{Z}^*$-flow are characterized. It is also shown that every $k$-regular graph, with $k$ either odd or congruent to 2 modulo 4, admits a $1$-sum $\{-1, 0, 1\}$-flow.

In this paper, we give an explicit and algorithmic description of Graver basis for the toric ideal associated with a simple undirected graph and apply the basis for testing the beta model of random graphs by Markov chain Monte Carlo method.

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n).At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. Let p be a prime. In this paper, a class of semisymmetric graphs of order 2 p 3 are determined. This work is a partial result for our long term goal to classify all semisymmetric graphs of order 2 p 3 .

Given a simple undirected graph, the problem of finding a maximum subset of vertices satisfying a nontrivial , interesting property that is hereditary on induced subgraphs , is known to be NP-hard....

Characterized are all simple undirected graphs G such that any real symmetric matrix that has graph G has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two ‘parallel’ induced paths. The remaining graphs that guarantee maximum multiplicity 2 are composed of certain identified families of ‘exceptional’ partial 2-trees that are not linear.

Let G and H be two simple undirected graphs. An embedding of the graph G in the graph H is an injective mapping f from the vertices of G into the vertices of H together with a mapping Pf of edges of G into paths in H. The dilation of the embedding is the maximum taken over all the lengths of the paths Pf(x,y) associated with the edges (x,y) of G.One challenge pointed out in [9] is to find embeddings of the de Bruijn graph in the hypercube of the same order which have a low dilation. For a de Bruijn graph of diameter D we give an embedding in a hypercube of the same diameter of dilation 2[D/51], and determine the edge-congestion and vertex-congestion of this embedding. Similar results are given for the shuffle-exchange graphs.

Exact algorithms for the minimum cost vertex blocker clique problem