Complement Of Graph Research Articles

Mathematical Properties on the Hyperbolicity of Interval Graphs

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.

Tight Nordhaus–Gaddum-Type Upper Bound for Total-Rainbow Connection Number of Graphs

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-colored graph is total-rainbow connected if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph G, the total-rainbow connection number of G, denoted by trc(G), is the minimum number of colors required in a total-coloring of G to make G total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus–Gaddum-type upper bound for the total-rainbow connection number. We prove that if G and $$\overline{G}$$ are connected complementary graphs on n vertices, then $$trc(G)+trc(\overline{G})\le 2n$$ when $$n\ge 6$$ and $$trc(G)+trc(\overline{G})\le 2n+1$$ when $$n=5$$ . Examples are given to show that the upper bounds are sharp for $$n\ge 5$$ . This completely solves a conjecture in Ma (Res Math 70(1–2):173–182, 2016).

Right-angled Artin groups and full subgraphs of graphs

For a finite graph [Formula: see text], let [Formula: see text] be the right-angled Artin group defined by the complement graph of [Formula: see text]. We show that, for any linear forest [Formula: see text] and any finite graph [Formula: see text], [Formula: see text] can be embedded into [Formula: see text] if and only if [Formula: see text] can be realized as a full subgraph of [Formula: see text]. We also prove that if we drop the assumption that [Formula: see text] is a linear forest, then the above assertion does not hold, namely, for any finite graph [Formula: see text], which is not a linear forest, there exists a finite graph [Formula: see text] such that [Formula: see text] can be embedded into [Formula: see text], though [Formula: see text] cannot be embedded into [Formula: see text] as a full subgraph.

LCD codes from adjacency matrices of graphs

It is shown how LCD codes with a particularly useful feature can be found from row spans over finite fields of adjacency matrices of graphs by considering these together with the codes from the associated reflexive graphs and complementary graphs. Application is made to some particular classes, including uniform subset graphs and strongly regular graphs where, if a p-ary code from a graph has this special LCD feature, the dimension can be found from the multiplicities modulo p of the eigenvalues of an adjacency matrix and, bounds on the minimum weight of the code and the dual code follow from the valency of the graph.

Landscape Metrics: Past Progress and Future Directions

This paper reviews progress in the field of landscape ecology related to the development of landscape metrics (i.e., spatial pattern indices). We first review the major formative historical developments that contributed to the coalescence of landscape metrics as a sub-field of landscape ecology and then examine recent literature highlighting several shortcomings related to their utility for understanding ecological processes and discuss several alternative approaches. Recent research recognizes some limitations of the patch-mosaic model (PMM), including the landscape metrics based on it, for capturing landscape heterogeneity and measuring functionality. Collapsing land cover information into nominal classes complicates identification of ecologically meaningful relationships and effective management. We explore several alternative methods for capturing landscape functionality and spatial heterogeneity including graph-based networks and gradient surface models with associated surface metrics. With complementary patch-based, gradient, and graph network models available, the goal for landscape ecologists is to select the correct approach, or combination of approaches, for investigating the issue at hand. Biases associated with the modifiable areal unit problem (MAUP) and its connection to heterogeneity and scale—both grain and extent—complicate these decisions, but empirical tools from spatial allometry may improve the ability for landscape ecologists to assess where metrics are capturing ecological processes versus the scale-dependency of the metrics themselves.

Nordhaus–Gaddum-Type Theorem for Total-Proper Connection Number of Graphs

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A path P in a total-colored graph G is called a total-proper path if (1) any two adjacent edges of P are assigned distinct colors; (2) any two adjacent internal vertices of P are assigned distinct colors; and (3) any internal vertex of P is assigned a distinct color from its incident edges of P. The total-colored graph G is total-proper connected if any two distinct vertices of G are connected by a total-proper path. The total-proper connection number of a connected graph G, denoted by tpc(G), is the minimum number of colors that are required to make G total-proper connected. In this paper, we first characterize the graphs G on n vertices with $$tpc(G)=n-1$$ . Based on this, we obtain a Nordhaus–Gaddum-type result for total-proper connection number. We prove that if G and $$\overline{G}$$ are connected complementary graphs on n vertices, then $$6\le tpc(G)+tpc(\overline{G})\le n+2$$ . Examples are given to show that the lower bound is sharp for $$n\ge 4$$ . The upper bound is reached for $$n\ge 4$$ if and only if G or $$\overline{G}$$ is the tree with maximum degree $$n-2$$ .

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

An embedding i mapsto p_iin mathbb {R}^d of the vertices of a graph G is called universally completable if the following holds: For any other embedding imapsto q_i~in mathbb {R}^{k} satisfying q_i^{T}q_j = p_i^{T}p_j for i = j and i adjacent to j, there exists an isometry mapping q_i to p_i for all iin V(G). The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on mathbb {Z}_2^n (n le 5) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.

Network Exploration by Complements of Graphs with Graph Coloring

Network Exploration by Complements of Graphs with Graph Coloring

Brain circuit–gene expression relationships and neuroplasticity of multisensory cortices in blind children

Sensory deprivation reorganizes neurocircuits in the human brain. The biological basis of such neuroplastic adaptations remains elusive. In this study, we applied two complementary graph theory-based functional connectivity analyses, one to evaluate whole-brain functional connectivity relationships and the second to specifically delineate distributed network connectivity profiles downstream of primary sensory cortices, to investigate neural reorganization in blind children compared with sighted controls. We also examined the relationship between connectivity changes and neuroplasticity-related gene expression profiles in the cerebral cortex. We observed that multisensory integration areas exhibited enhanced functional connectivity in blind children and that this reorganization was spatially associated with the transcription levels of specific members of the cAMP Response Element Binding protein gene family. Using systems-level analyses, this study advances our understanding of human neuroplasticity and its genetic underpinnings following sensory deprivation.

Self-inverse unicyclic graphs and strong reciprocal eigenvalue property

We consider only simple graphs. A graph G is said to be nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. An invertible graph G is said to be a self-inverse graph if G is isomorphic to its inverse. Let H be the class of connected bipartite graphs with unique perfect matchings. We consider the problem of characterizing the graphs in H which are self-inverse graphs. In this article, we answer this problem for the class of unicyclic graphs.The study of the strong reciprocal eigenvalue property is closely related to the study of graphs which are isomorphic to their inverse graphs. A nonsingular graph G is said to satisfy the property (SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. In this article, we show that if a unicyclic graph G in H has property (SR), then G is invertible and the inverse graph of G is unicyclic. As an application, we show that a noncorona unicyclic graph in H with property (SR) can have one of the five specified structures. Finally, our discussions lead to the following problem. Does there exist a unicyclic graph G∈H which has property (SR) but G is not self-inverse?

SEMIGLOBAL TOTAL DOMINATION IN GRAPHS

A subset D of vertices of a connected graph G is called a semiglobal total dominating set if D is a dominating set for G and G sc and has no isolated vertex in G, where G sc is the semi complementary graph of G. The semiglobal total domination number is the minimum cardinality of a semiglobal total dominating set of G and is denoted by γ sgt (G). In this paper exact values for γ sgt (G) are obtained for some graphs like cycles, wheel and paths are presented as well.

Inverse graphs associated with finite groups

Let $(\Gamma,*)$ be a finite group and $S$ a possibly empty subset of $\Gamma$ containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with $\Gamma$ whose set of vertices coincides with $\Gamma$ such that two distinct vertices $u$ and $v$ are adjacent if and only if either $u * v\in S$ or $v * u\in S$. We then investigate its algebraic and combinatorial structures.

Concepts of Interval-Valued Neutrosophic Graphs

Broumi et al. [15] proposedthe concept of interval-valued neutrosophic graphs. In this research article, we first show that there are some flaws in Broumi et al. [15] ’s definition which violates condition of an interval. We then modify the definition of an interval-valued neutrosophic graph. We present some operations on interval-valued neutrosophic graphs. We discuss the concepts of self-complementary and self weak complementary interval-valued neutrosophic complete graphs. We also describe regularity of interval-valued neutrosophic graphs.

The least signless Laplacian eignvalue of the complements of unicyclic graphs

Let Snc be the set of all connected graph each of which is a complement of an n-vertex unicyclic graph. Li and Wang (2012) determined the graphs with the least signless Laplacian eignvalue among all the graphs of the complements of n-vertex trees. In this paper, as a continuance of it, the unique graph among Snc which minimizes the least signless Laplacian eigenvalue is identified.

Non-edge orientation and vertex ordering characterizations of some classes of bigraphs

Chordal graphs, permutation graphs, and interval graphs are among many classes of graphs which can be characterized by the existence of certain acyclic orientations and vertex orderings. These types of characterizations exist for some of their bipartite analogues such as chordal bipartite graphs and bipartite permutation graphs. Chvátal proved that a bipartite graph G is chordal bipartite if and only if the complement G¯ of G has a vertex ordering ≺ such that for every induced path abcd in G¯, a≺b implies c≺d. Recently, Le proved that a bipartite graph G is a permutation graph if and only if G¯ admits an acyclic orientation such that for every induced path abcd in G¯, ab is an oriented edge if and only if cd is. Interestingly these orientation and vertex ordering characterizations are stated on the complements of bipartite graphs. We show that interval bigraphs and interval containment bigraphs also admit similar characterizations in terms of vertex orderings and acyclic orientations of their complements.

On the zero-divisor graph of Rickart *-rings

In this paper, we study the zero-divisor graph of Rickart *-rings. We determine the condition on Rickart *-ring so that its zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices form a complete subgraph. We characterize Rickart *-rings for which the complement of the zero-divisor graph is connected. The diameter and girth of these graphs are characterized. Further, for a *-ring [Formula: see text] with unity we associate a graph, [Formula: see text], having all nonzero elements of [Formula: see text] as vertices and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text] is a unit in [Formula: see text]. For a finite Rickart *-ring [Formula: see text] it is proved that [Formula: see text] is connected if and only if [Formula: see text] is not isomorphic to [Formula: see text] or [Formula: see text] (for any [Formula: see text].

Using k-Mix-Neighborhood Subdigraphs to Compute Canonical Labelings of Digraphs

This paper presents a novel theory and method to calculate the canonical labelings of digraphs whose definition is entirely different from the traditional definition of Nauty. It indicates the mutual relationships that exist between the canonical labeling of a digraph and the canonical labeling of its complement graph. It systematically examines the link between computing the canonical labeling of a digraph and the k-neighborhood and k-mix-neighborhood subdigraphs. To facilitate the presentation, it introduces several concepts including mix diffusion outdegree sequence and entire mix diffusion outdegree sequences. For each node in a digraph G, it assigns an attribute m_NearestNode to enhance the accuracy of calculating canonical labeling. The four theorems proved here demonstrate how to determine the first nodes added into M a x Q ( G ) . Further, the other two theorems stated below deal with identifying the second nodes added into M a x Q ( G ) . When computing C m a x ( G ) , if M a x Q ( G ) already contains the first i vertices u 1 , u 2 , ⋯ , u i , Diffusion Theorem provides a guideline on how to choose the subsequent node of M a x Q ( G ) . Besides, the Mix Diffusion Theorem shows that the selection of the ( i + 1 ) th vertex of M a x Q ( G ) for computing C m a x ( G ) is from the open mix-neighborhood subdigraph N + + ( Q ) of the nodes set Q = { u 1 , u 2 , ⋯ , u i } . It also offers two theorems to calculate the C m a x ( G ) of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate M a x Q ( G ) of a digraph. Through software testing, the correctness of our algorithms is preliminarily verified. Our method can be utilized to mine the frequent subdigraph. We also guess that if there exists a vertex v ∈ S + ( G ) satisfying conditions C m a x ( G − v ) ⩽ C m a x ( G − w ) for each w ∈ S + ( G ) ∧ w ≠ v , then u 1 = v for M a x Q ( G ) .

Abstract WMP95: A Transition of Care Program to Reduce Stroke Related Hospital Readmissions

Introduction: Readmission to the hospital after discharge following a stroke or TIA remains a nation-wide problem. While the CMS national benchmark was approximately 12% in 2015, our hospital Medicare stroke readmission rate rose from approximately 12% at the end of 2014 to 28.6% in February 2015. Our goal was a reduction in stroke readmission rates to below the national benchmark of 12% by December 2015. Hypothesis: We hypothesized that implementing a transition of care program at our 200 bed community hospital would reduce hospital stroke-related readmissions. Methods: In March 2015, a random sample of forty stroke/TIA patients that were discharged home between December of 2014 and February of 2015 were interviewed. The patients were asked about barriers to discharge, what could have improved the discharge experience, and what problems they encountered that could have resulted in a readmission. Based on their answers, risk factors were identified using an inverse Pareto graph and a transition of care program was implemented which included the following work flow: 1) daily rounding to query patients regarding insight into stroke risk factors, environmental concerns, and social impacts to discharge in the stroke unit by the stroke coordinator (a registered nurse); 2) a discharge telephone call within two business days to high risk patients identified during rounds focusing on review of the discharge summary, re-education regarding stroke risk factors, and ensuring that follow-up appointments were in place; 3) an outpatient follow-up appointment with a board certified vascular neurologist within two weeks of discharge. Results: Our transition of care program resulted in an improvement of 82.5%, with a Medicare stroke re-admission rate of 5% in December 2015. As of May 2016, our year-to-date hospital stroke readmission rate is 8.1%, while the current CMS national average is 12.7%. Conclusions: A transition of care program is implementable in a community hospital setting, and results in reduced stroke-related hospital readmissions. Its success emphasizes the importance of identifying high risk patients and assessing individual drivers of readmission risk.

NEW CONCEPTS OF REGULAR INTERVAL-VALUED FUZZY GRAPHS

Recently, interval-valued fuzzy graph is a growing research topic as it is the generalization of fuzzy graphs. The interval-valued fuzzy graphs are more flexible and compatible than fuzzy graphs due to the fact that they allowed the degree of membership of a vertex to an edge to be represented by interval values in [0.1] rather than the crisp values between 0 and 1. In this paper, we introduce the concepts of regular and totally regular interval-valued fuzzy graphs and discusses some properties of the <TEX>${\mu}$</TEX>-complement of interval-valued fuzzy graph. Self <TEX>${\mu}$</TEX>-complementary interval-valued fuzzy graphs and self-weak <TEX>${\mu}$</TEX>-complementary interval-valued fuzzy graphs are defined and a necessary condition for an interval valued fuzzy graph to be self <TEX>${\mu}$</TEX>-complementary is discussed. We define busy vertices and free vertices in interval valued fuzzy graph and study their image under an isomorphism.

Computing a Clique Tree with the Algorithm Maximal Label Search

The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.