Several Families Of Graphs Research Articles

Generalized Splines on Graphs with Two Labels and Polynomial Splines on Cycles

Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley-Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet.
 This paper focuses on the theory of generalized splines. A generalized spline on a graph $G$ with each edge labeled by an ideal in a ring $R$ consists of a vertex-labeling by elements of $R$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $R$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $\mathbb{C}[x,y]$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines.

Several properties of antiadjacency matrices of directed graphs

<p>Let $ G $ be a directed graph with $ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\; n. $ The adjacency matrix of the directed graph $ G $ is a matrix $ A = \left[{a}_{ij}\right] $ of order $ n\times n, $ such that for $ i\ne j $, if there is an arc from $ i $ to $ j, $ then $ {a}_{ij} = 1 $, otherwise $ {a}_{ij} = 0 $. Matrix $ B = J-A $ is called the antiadjacency matrix of the directed graph $ G, $ where $ J $ is the matrix of order $ n\times n $ with all of those entries are one. In this paper, we provided several properties of the adjacency matrices of directed graphs, such as a determinant of a directed graphs, the characteristic polynomial of acyclic directed graphs, and regular directed graphs. Moreover, we discuss antiadjacency energy of acyclic directed graphs and give some examples of antiadjacency energy for several families of graphs.</p>

On local distance antimagic labeling of graphs

Let G = ( V , E ) be a graph of order n and let f : V → { 1 , 2 … , n } be a bijection. For every vertex v ∈ V , we define the weight of the vertex v as w ( v ) = ∑ x ∈ N ( v ) f ( x ) where N(v) is the open neighborhood of the vertex v. The bijection f is said to be a local distance antimagic labeling of G if w ( u ) ≠ w ( v ) for every pair of adjacent vertices u , v ∈ V . The local distance antimagic labeling f defines a proper vertex coloring of the graph G, where the vertex v is assigned the color w(v). We define the local distance antimagic chromatic number χ l d ( G ) to be the minimum number of colors taken over all colorings induced by local distance antimagic labelings of G. In this paper we obtain the local distance antimagic labelings for several families of graphs including the path Pn , the cycle Cn , the wheel graph Wn , friendship graph Fn , the corona product of graphs G ° K m ¯ , complete multipartite graph and some special types of the caterpillars. We also find upper bounds for the local distance antimagic chromatic number for these families of graphs.

Graphs with small distance-based complexities

Suppose G is a connected graph. The Wiener complexity (or briefly W-complexity) CW(G) is the number of different contributions to Wiener index in its summation formula. Also, the Szeged complexity (or briefly Sz-complexity) CSz(G) can be defined similarly. The main goal of the current work is to investigate graphs with small complexity. We classify all unicycle graphs, all distance-balanced graphs, and all regular graphs of diameter 2 with Sz-complexity one. In this way, the Sz-complexity and W-complexity of several families of graphs are determined.

Fixing Numbers of Point-Block Incidence Graphs

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in the graph. Fixing numbers were first introduced by Laison and Gibbons, and independently by Erwin and Harary. Fixing numbers have also been referred to as determining numbers by Boutin. The main motivation is to remove all symmetries from a graph. A very simple application is in the creation of QR codes where the symbols must be fixed against any rotation. We determine the fixing number for several families of graphs, including those arising from combinatorial block designs. We also present several infinite families of graphs with an even stronger condition, where fixing any vertex in a graph fixes every vertex.

Kemeny's constant for nonbacktracking random walks

AbstractKemeny's constant for a connected graph is the expected time for a random walk to reach a randomly chosen vertex , regardless of the choice of the initial vertex. We extend the definition of Kemeny's constant to nonbacktracking random walks and compare it to Kemeny's constant for simple random walks. We explore the relationship between these two parameters for several families of graphs and provide closed‐form expressions for regular and biregular graphs. In nearly all cases, the nonbacktracking variant yields the smaller Kemeny's constant.

Curvature on graphs via equilibrium measures

AbstractWe introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by have diameter bounded by (a Bonnet–Myers theorem), that implies that has constant curvature (a Cheng theorem) and that there is a spectral gap (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.

The Reidemeister spectrum of 2-step nilpotent groups determined by graphs

ABSTRACT In this paper we study the Reidemeister spectrum of finitely generated torsion-free 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.

K-Product cordial labeling of product of graphs

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let [Formula: see text] be a map from [Formula: see text] to [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. For each edge [Formula: see text] assign the label [Formula: see text] [Formula: see text]. [Formula: see text] is called a k-product cordial labeling if [Formula: see text], and [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] denote the number of vertices and edges, respectively, labeled with [Formula: see text] [Formula: see text]. A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling.

A path forward: Tropicalization in extremal combinatorics

Many important problems in extremal combinatorics can be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom(H1,G)a1⋯hom(Hk,G)ak≥hom(Hk+1,G)ak+1⋯hom(Hm,G)am holds for some fixed graphs H1,…,Hm and all graphs G. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs U={H1,…,Hm}, the exponent vectors of valid pure binomial inequalities in graphs of U form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman ([17]), we show that the cone of valid inequalities is indeed rational polyhedral when all graphs Hi are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection U. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erdős-Simonovits conjecture recently proved in its original form in [29] and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry, first applied in extremal combinatorics in [3]. We prove several results about tropicalizations which may be of independent interest.

The Equidistant Dimension of Graphs

A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x, y in V (G) {setminus } S there is a vertex w in S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.

Two extensions of Leech labeling to the class of all graphs

Let be a tree of order n and let be an injective edge labeling of T. The weight of a path P is the sum of the labels of the edges of P and is denoted by If the set of weights of the paths in T is then f is called a Leech labeling of T and a tree which admits a Leech labeling is called a Leech tree. In this paper, we present two extensions of Leech labeling to the class of all graphs, namely, Leech labeling and geodesic Leech labeling. For this purpose we introduce two parameters and where is the total number of paths in G and is the total number of geodesic paths in G. We determine the values of these parameters for some graph classes. We present several families of graphs which are Leech graphs, geodesic Leech graphs and also graphs which are not Leech graphs. A list of open problems is also included.

On the double Roman bondage numbers of graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite graphs.

A new lower bound on graph gonality

We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone, but it is subgraph monotone and invariant under subdivision. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.

Improper interval edge colorings of graphs

A k-improper edge coloring of a graph G is a mapping α:E(G)⟶N such that at most k edges of G with a common endpoint have the same color. An improper edge coloring of a graph G is called an improper interval edge coloring if the colors of the edges incident to each vertex of G form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph G defined as the smallest k such that G has a k-improper interval edge coloring; we denote the smallest such k by μint(G). We prove upper bounds on μint(G) for general graphs G and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine μint(G) exactly for G belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer k, there exists a graph G with μint(G)=k. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring.

LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS

In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$. A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balanced with a closed neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number $k$ ($k\geq 2$), for which a graph $G$ has a locally-balanced $k$-partition with an open (a closed) neighborhood, is called an $lb$-open ($lb$-closed) chromatic number of $G$ and denoted by $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the $lb$-open and $lb$-closed chromatic numbers of several families of graphs. We also consider the connections of $lb$-open and $lb$-closed chromatic numbers of graphs with other chromatic numbers such as injective and $2$-distance chromatic numbers.

The toughness of Kneser graphs

The toughnesst(G) of a graph G is a measure of its connectivity that is closely related to Hamiltonicity. Xiaofeng Gu, confirming a longstanding conjecture of Brouwer, recently proved the lower bound t(G)≥ℓ/λ−1 on the toughness of any connected ℓ-regular graph, where λ is the largest nontrivial absolute eigenvalue of the adjacency matrix. Brouwer had also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly ℓ/λ. Cioabă and Wong confirmed Brouwer's observation for several families of graphs, including Kneser graphs K(n,2) and their complements, with the exception of the Petersen graph K(5,2). In this paper, we extend these results and determine the toughness of Kneser graphs K(n,k) when k∈{3,4} and n≥2k+1 as well as for k≥5 and sufficiently large n (in terms of k). In all these cases, the toughness is attained by the complement of a maximum independent set and we conjecture that this is the case for any k≥5 and n≥2k+1.

Comparing Wiener complexity with eccentric complexity

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices in G. The Wiener complexity of G is the number of different transmissions of its vertices. Similarly, the eccentric complexity of G is defined as the number of different eccentricities of its vertices. In this paper these two complexities are compared. The complexities are first studied on Cartesian product graphs. Transmission indivisible graphs and arithmetic transmission graphs are introduced to demonstrate sharpness of upper and lower bounds on the Wiener complexity, respectively. It is shown that for almost all graphs the Wiener complexity is not smaller than the eccentric complexity. This property is proved for trees, the equality holding precisely for center-regular trees. Several families of graphs in which the complexities are equal are constructed. Using the Cartesian product, it is proved that the eccentric complexity can be arbitrarily larger than the Wiener complexity. Additional infinite families of graphs with this property are constructed by amalgamating universally diametrical graphs with center-regular trees.

The Noncrossing Bond Poset of a Graph

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$.
 Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

Graphs with equal domination and independent domination numbers

Let γ(G) and i(G) denote the domination number and independent domination number of a graph G. In this article, we establish a sufficient condition for a graph G to satisfy which yields some of the well known classical theorems as corollaries. Further, we also construct several families of graphs G satisfying γ(G) = i(G) using the sufficient condition.