Disjoint Graphs Research Articles

Network of co-expressed circadian genes, childhood maltreatment and sleep quality in bipolar disorders

ABSTRACT Bipolar disorder (BD) is a chronic and burdensome psychiatric disease, characterized by variations in mood and energy. The literature has consistently demonstrated an association between BD and childhood maltreatment (CM), and genetic variants of circadian genes have been associated with an increased vulnerability to develop BD. In this context, environmental factors such as CM may also contribute to the susceptibility to BD through alterations in the functioning of the biological clock linked to modifications of expression of circadian genes. In this study, we explored the associations between childhood maltreatment, sleep quality, and the level of expression of a comprehensive set of circadian genes in lymphoblastoid cell lines from patients with BD. The sample consisted of 52 Caucasian euthymic patients with a diagnosis of BD type 1 or type 2. The exposure to CM was assessed with the Childhood Trauma Questionnaire (CTQ), and the sleep quality was assessed using the Pittsburgh Sleep Quality Index. We measured the expression of 18 circadian genes using quantitative RT-PCR: ARNTL2, BHLHE40, BHLHE41, CLOCK, CRY1, CRY2, CSNK1D, CSNK1E, DBP, GSK3B, NPAS2, NR1D1, PER1, PER2, PER3, PPARGC1A, RORA, and RORB. Gene expression networks were analyzed with the disjoint graphs method. Compared to the other investigated transcripts, PPARGC1A was the only one whose expression level was differentially affected in patients who have experienced CM and, more specifically, physical abuse. We observed no significant effects of the other CTQ subscores (emotional and sexual abuses, physical and emotional neglects), nor of the sleep quality on the network of circadian genes expression. Although requiring replication in larger cohorts, the result obtained here is consistent with the hypothesis of an influence of CM exposure on circadian systems and highlights the importance of PPARGC1A in these processes.

Hasse diagrams with large chromatic number

For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( n 3 / 4 ) . Such graphs have chromatic number Ω ( n 1 / 4 ) , which significantly improves the previously best-known constructions of Hasse diagrams having chromatic number Θ ( log n ) . In addition, if we also require girth of at least k ⩾ 5 , we construct such Hasse diagrams with independence number at most n 1 − 1 2 k − 4 + o ( 1 ) . The proofs are based on the existence of point-line arrangements in the plane with many incidences and avoids certain forbidden subconfigurations, which we find of independent interest. These results also have the following surprising geometric consequence. They imply the existence of a family C of n curves in the plane such that the disjointness graph G of C is triangle-free (or has high girth), but the chromatic number of G is polynomial in n. Again, the previously best-known construction, due to Pach, Tardos and Tóth, had only logarithmic chromatic number.

Disjointness graphs of segments in the space

AbstractThe disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$, $$d \ge 2$$, is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$, where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.

On the chromatic number of disjointness graphs of curves

Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G)=k, then χ(G)≤(k+12). If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G)≤k+12(k+23). Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor.

The distance Laplacian and distance signless Laplacian spectrum of the subdivision-vertex join and subdivision-edge join of two regular graphs

Let [Formula: see text] be a connected graph with a distance matrix [Formula: see text]. Let [Formula: see text] and [Formula: see text] be, respectively, the distance Laplacian matrix and the distance signless Laplacian matrix of graph [Formula: see text], where [Formula: see text] denotes the diagonal matrix of the vertex transmissions in [Formula: see text]. The eigenvalues of [Formula: see text] and [Formula: see text] constitute the distance Laplacian spectrum and distance signless Laplacian spectrum, respectively. The subdivision graph [Formula: see text] of a graph [Formula: see text] is obtained by inserting a new vertex into every edge of [Formula: see text]. We denote the set of such new vertices by [Formula: see text]. The subdivision-vertex join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. The subdivision-edge join of two vertex disjoint graphs [Formula: see text] and [Formula: see text] denoted by [Formula: see text], is the graph obtained from [Formula: see text] and [Formula: see text] by joining each vertex of [Formula: see text] with every vertex of [Formula: see text]. In this paper, we determine the distance Laplacian and distance signless Laplacian spectra of subdivision-vertex join and subdivision-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. As an application we exhibit some infinite families of cospectral graphs and find the respective spectra of the Jahangir graph [Formula: see text].

Resistance Distance in H-Join of Graphs G1,G2,…,Gk

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , … , G k . Recall that H is an arbitrary graph with V ( H ) = { 1 , 2 , … , k } , and G 1 , G 2 , … , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , … , G k , denoted by ⋁ H { G 1 , G 2 , … , G k } , is a graph formed by taking G 1 , G 2 , … , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , … , G k } , and then give a { 1 } -inverse L ( ⋁ H { G 1 , G 2 , … , G k } ) { 1 } or group inverse L ( ⋁ H { G 1 , G 2 , … , G k } ) # of L ( ⋁ H { G 1 , G 2 , … , G k } ) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , … , G k } . In addition, some applications are presented in this paper.

Genus polynomials and crosscap‐number polynomials for ring‐like graphs

AbstractAn H‐linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H‐ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H‐ring‐like graphs from sequences of H‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.

The change in multiplicity of an eigenvalue due to adding or removing edges

We investigate the change in the multiplicities of the eigenvalues of a Hermitian matrix with a simple graph G, when edges are inserted into G or removed from G. We focus upon cases in which the multiplicity of the eigenvalue does not change due to inserting or removing edges incident to a vertex. Furthermore, we show how the change in the multiplicities of the eigenvalues occur, when two disjoint graphs are connected with one edge, based upon the status of the vertices that are connected. Lastly, we give the possible classifications of cut-edges in a graph and characterize the occurrence of each possible status.

Distance matrices on the H-join of graphs: A general result and applications

Given a graph H with vertices 1,…,s and a set of pairwise vertex disjoint graphs G1,…,Gs, the vertex i of H is assigned to Gi. Let G be the graph obtained from the graphs G1,…,Gs and the edges connecting each vertex of Gi with all the vertices of Gj for all edge ij of H. The graph G is called the H-join of G1,…,Gs. Let M(G) be a matrix on a graph G. A general result on the eigenvalues of M(G), when the all ones vector is an eigenvector of M(Gi) for i=1,2,…,s, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G1,…,Gs are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.

On Graded Betti Numbers of a Class of Graphs

In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.

Spectra of Subdivision-Vertex Join and Subdivision-Edge Join of Two Graphs

The subdivision graph $\mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The \emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1\dot{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The \emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1\underline{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $\mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.

The {1}-inverse of the Laplacian of subdivision-vertex and subdivision-edge coronae with applications

The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let and be two vertex disjoint graphs. The subdivision-vertex corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of . The subdivision-edge corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of , where is the set of inserted vertices of . In this paper, -inverse for the Laplacian matrices of graphs and are proposed, based on which the explicit resistance distance can be obtained for the two-vertex resistance between arbitrary vertices in the graphs.

Entity resolution in disjoint graphs: An application on genealogical data

Entity Resolution (ER) is the process of identifying references referring to the same entity from one or more data sources. In the ER process, most existing approaches exploit the content information of references, categorized as content-based ER, or additionally consider linkage information among references, categorized as context-based ER. However, in new applications of ER, such as in the genealogical domain, the very limited linkage information among references results in a disjoint graph in which the existing content-/context-based ER techniques have very limited applicability. Therefore, in this paper we propose first, to use the homophily principle for augmentation of the original input graph by connecting the potential similar references, and second, to use a Random Walk based approach to consider contextual information available for each reference in the augmented graph. We evaluate the proposed method by applying it to a large genealogical dataset and we succeed to predict 420,000 reference matches with precision 92% and discover six novel and informative patterns among them which can not be detected in the original disjoint graph.

Spectra of some Operations on Graphs

In this paper, we consider a finite undirected and connected simple graph G(E, V) with vertex set V(G) and edge set E(G).We introduced a new computes the spectra of some operations on simple graphs [union of disjoint graphs, join of graphs, cartesian product of graphs, strong cartesian product of graphs, direct product of graphs].

Even harmonious labelings of disjoint graphs with a small component

A graph G with q edges is said to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y)(modq), the resulting edge labels are distinct. If G is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2(q−1) and the induced function f∗ from the edges of G to 0,2,…,2(q−1) defined by f∗(xy)=f(x)+f(y)(mod2q) is bijective. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C3,C4,K4 or W4 as a component.

Cover Product and Betti Polynomial of Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

Laplacian spectral characterization of some graph join

For two disjoint graphs G and H, the join of G and H, denoted by G ∨ H, is the graph obtained from G ∪ H by joining each vertex of G to each vertex of H. A graph is said to be DLS if there is no other non-isomorphic graph with the same Laplacian spectrum. For a connected DLS graph G with a cut vertex, we prove that G ∨ Kr is DLS, where Kr is a complete graph. For a disconnected DLS graph G with n ⩾ 10 vertices and m ⩽ n — 4 edges, we show that G ∨ (Kr — e) is DLS, where Kr — e is the graph obtained by deleting one edge of Kr. Applying these results we can obtain new DLS graphs.

Resistance distance and Kirchhoff index of [formula omitted]-vertex join and [formula omitted]-edge join of two graphs

Let G=(V(G),E(G)) be a graph with vertex set V(G) and edge set E(G). The R-graph of a graph G, denoted by R(G), is the graph obtained from G by adding a vertex ve and then joining ve to the end vertices of e for each e∈E(G). Let G1 and G2 be two vertex disjoint graphs. The R-vertex join of G1 and G2, denoted by G1〈v〉G2, is the graph obtained from R(G1) and G2 by joining every vertex of V(G1) with every vertex of V(G2). The R-edge join of G1 and G2, denoted by G1〈e〉G2, is the graph obtained from R(G1) and G2 by joining every vertex of I(G1) with every vertex of V(G2), where I(G1) is the set of the added vertices of R(G1). In this paper, we formulate the resistance distances and the Kirchhoff index of G1〈v〉G2 and G1〈e〉G2 respectively.

Signless Laplacian spectral characterization of some joins

The join of two disjoint graphs G and H, denoted by G ∨ H, is the graph obtained by joining each vertex of G to each vertex of H. In this paper, the signless Laplacian characteristic polynomial of the join of two graphs is first formulated. And then, a lower bound for the i-th largest signless Laplacian eigenvalue of a graph is given. Finally, it is proved that G ∨ K_m, where G is an (n − 2)-regular graph on n vertices, and K_n ∨ K_2 except for n = 3, are determined by their signless Laplacian spectra.

Critical Graphs in Index Coding

In this paper, we define critical graphs as minimal graphs that support a given set of rates for the index coding problem and study them for both the one-shot and asymptotic setups. For the case of equal rates, we find the critical graph with minimum number of edges for both one-shot and asymptotic cases. For the general case of possibly distinct rates, we show that for one-shot and asymptotic linear index coding, as well as asymptotic nonlinear index coding, each critical graph is a union of disjoint strongly connected subgraphs. On the other hand, we identify a non-USCS critical graph for a one-shot nonlinear index coding problem. Next, we identify a few graph structures that are critical. In addition, we show that the capacity region of the index coding is additive for union of disjoint graphs.