Twin Vertices Research Articles

Fractional revival between twin vertices

In this paper, we provide a characterization of fractional revival between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we characterize fractional revival between apexes of double cones.

Strong cospectrality and twin vertices in weighted graphs

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also determine when strong cospectrality is preserved under Cartesian and direct products of graphs. Moreover, we generalize known results about equitable and almost equitable partitions and use these to determine which joins of the form $X\vee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.

Laplacian state transfer on graphs with an edge perturbation between twin vertices

We consider quantum state transfer relative to the Laplacian matrix of a graph. Let N(u) denote the set of all neighbors of a vertex u in a graph G. A pair of vertices u and v are called twin vertices of G provided N(u)∖{v}=N(v)∖{u}. We investigate the existence of quantum state transfer between a pair of twin vertices in a graph when the edge between the vertices is perturbed. We find that removal of any set of pairwise non-adjacent edges from a complete graph with a number of vertices divisible by 4 results Laplacian perfect state transfer (or LPST) at π2 between the end vertices of every edge removed. Further, we show that all Laplacian integral graphs with a pair of twin vertices exhibit LPST when the edge between the vertices is perturbed. In contrast, we conclude that LPST can be achieved in every complete graph between the end vertices of any number of suitably perturbed non-adjacent edges. The results are further generalized to obtain a family of edge perturbed circulant graphs exhibiting Laplacian pretty good state transfer (or LPGST) between twin vertices. A subfamily of which is also identified to admit LPST at π2.

On the maximum cardinality cut problem in proper interval graphs and related graph classes

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this work we consider the parameterized complexity of this problem. We show that the maximum cardinality cut problem in proper/unit interval graphs is FPT when parameterized by the maximum number of non-empty bubbles in a column of its bubble model. We then generalize this result to a more general graph class by defining new parameters related to the well-known clique-width parameter.Specifically, we define an (α,β,δ)-clique-width decomposition of a graph as a clique-width decomposition in which at each step the following invariant is preserved: after discarding at most δ labels, a) every label consists of at most β sets of twin vertices, and b) all the labels together induce a graph with independence number at most α. We show that for every two constants α,δ>0 the problem is FPT when parameterized by β plus the smallest width of an (α,β,δ)-clique-width decomposition.

Minimum Partition of an r − Independence System

Graph partitioning has been studied in the discipline between computer science and applied mathematics. It is a technique to distribute the whole graph data as a disjoint subset to a different device. The minimum graph partition problem with respect to an independence system of a graph has been studied in this paper. The considered independence system consists of one of the independent sets defined by Boutin. We solve the minimum partition problem in path graphs, cycle graphs, and wheel graphs. We supply a relation of twin vertices of a graph with its independence system. We see that a maximal independent set is not always a minimal set in some situations. We also provide realizations about the maximum cardinality of a minimum partition of the independence system. Furthermore, we study the comparison of the metric dimension problem of a graph with the minimum partition problem of that graph.

Bounds for metric dimensions of generalized neighborhood corona graphs

In this paper, the authors analysed metric dimensions of arbitrary graphs G★˜∧i=1|V(G)|Hi in which graphs G,H1,H2,…,H|V(G)| are non-trivial, G is connected, and ★˜ denotes generalized neighborhood corona operation. We found lower bounds of dim(G★˜∧i=1|V(G)|Hi) as function of dimA(Hi) where dimA(Hi) denotes adjacency metric dimensions of Hi. We also found upper bounds of dim(G★˜∧i=1|V(G)|Hi) when G does not contain pair of false twin vertices. Furthermore, we found a characteristic of dim(G★˜∧i=1|V(G)|Hi) which indicates that our lower bounds are strict.

The normalized distance Laplacian

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

Determining number and cost of generalized Mycielskian graphs

A set $S$ of vertices is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called its determining number, $Det(G)$. A graph $G$ is said to be $d$-distinguishable if there is a coloring of the vertices with $d$ colors so that only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number, $Dist(G)$. If $Dist(G) = 2$, the cost of 2-distinguishing, $\rho(G)$, is the size of a smallest color class over all 2-distinguishing colorings of $G$. The Mycielskian, $\mu(G)$, of a graph $G$ is constructed by adding a shadow master vertex $w$, and for each vertex $v_i$ of $G$ adding a shadow vertex $u_i$ with edges so that the neighborhood of $u_i$ in $\mu(G)$ is the same as the neighborhood of $v_i$ in $G$ with the addition of $w$. That is, $N(u_i)=N_G(v_i)\cup\{w\}$. The generalized Mycielskian $\mu^{(t)}(G)$ of a graph $G$ is a Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers above and below, and $w$ only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if $G \neq K_2$ is twin-free with no isolated vertices, then $Det(\mu^{(t)}(G)) = Det(G)$. Further, if $Det(G) = k \geq 2$ and $t \ge k-1$, then $Dist(\mu^{(t)}(G))=2$, and $Det(\mu^{(t)}(G)) = \rho(\mu^{(t)}(G))= k$. For $G$ with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which $Det(\mu^{(t)}(G)) = (t{+}1) Det(G)$.

Binomial incidence matrix of a semigraph

A semigraph, defined as a generalization of graph by Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum

Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it is isomorphic to each digraph to which it is cospectral. Convincing numerical evidence to support the claim that this property is extremely rare is provided. Nonetheless, the first infinite family of connected digraphs that is SHDS is constructed. This family is obtained via the introduction of twin vertices into a structure that is named negative tetrahedron. This special digraph, that exhibits extreme spectral behavior, is contained in the surprisingly small collection of all digraphs with exactly one negative eigenvalue, which is determined as an intermediate result.

On the displacement of eigenvalues when removing a twin vertex

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation.

Complete multipartite graphs and Braess edges

Given an irreducible stochastic matrix T, Kemeny's constant κ(T) measures the expected number of steps from any initial state to a randomly chosen final state, and is thus regarded as an indicator of the overall transit efficiency of the corresponding Markov chain. For a random walk on a connected graph with twin vertices, we present a formula for the change in Kemeny's constant after the insertion of an edge between the twins. Using that result, we investigate the circumstances under which inserting a new edge increases Kemeny's constant. In particular, we characterise the complete multipartite graphs without dominating vertices having the property that adding any new edge increases Kemeny's constant. We establish an analogous result for complete split graphs. We also prove that if a complete multipartite graph has enough dominating vertices, then there is always a non–edge whose addition will decrease Kemeny's constant.

The twin non-commuting graph of a group

In this paper, the twin non-commuting graph of the group G is introduced by partitioning the non-commuting graph vertices. This classification of the vertices based on special property, precisely being twin vertices property. We choose a vertex from each class as a representing vertex for the twin non-commuting graph. Moreover the adjacency of two vertices in twin non-commuting graph depends on the connectivity of them in the main non-commuting graph. We observe that the twin non-commuting graph of an AC-group is a complete graph. Moreover, some results about the metric dimension of the non-commuting graph is obtained. For instance, the metric dimension of the non-commuting graph is 3 if and only if it is associated with $$S_3,D_8,Q_8$$ .

Isotropic Matroids III: Connectivity

The isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of $M[IAS(G)]$, and if $G$ has at least four vertices, then $M[IAS(G)]$ is vertically 5-connected if and only if $G$ is prime (in the sense of Cunningham's split decomposition). We also show that $M[IAS(G)]$ is $3$-connected if and only if $G$ is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if $G$ has $n\geq7$ vertices then $M[IAS(G)]$ is not vertically $n$-connected. This abstract-seeming result is equivalent to the more concrete assertion that $G$ is locally equivalent to a graph with a vertex of degree $<\frac{n-1}{2}$.

Critical ideals of signed graphs with twin vertices

This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph G=(V,E) and d∈Z|V|, let Gd be the graph obtained from G by duplicating dv times or replicating −dv times the vertex v when dv>0 or dv<0, respectively. Moreover, given δ∈{0,1,−1}|V|, letTδ(G)={Gd:d∈Z|V| such that dv=0Tδ(G)={Gd:if and only if δv=0 and dvδv>0 otherwise} be the set of graphs sharing the same pattern of duplication or replication of vertices. More than one half of the critical ideals of a graph in Tδ(G) can be determined by the critical ideals of G. The algebraic co-rank of a graph G is the maximum integer i such that the i-th critical ideal of G is trivial. We show that the algebraic co-rank of any graph in Tδ(G) is equal to the algebraic co-rank of Gδ. Moreover, the algebraic co-rank can be determined by a simple evaluation of the critical ideals of G. For a large enough d∈ZV(G), we show that the critical ideals of Gd have similar behavior to the critical ideals of the disjoint union of G and some set {Knv}{v∈V(G)|dv<0} of complete graphs and some set {Tnv}{v∈V(G)|dv>0} of trivial graphs. Additionally, we pose important conjectures on the distribution of the algebraic co-rank of the graphs with twins vertices. These conjectures imply that twin-free graphs have a large algebraic co-rank, meanwhile a graph having small algebraic co-rank has at least one pair of twin vertices.

Splitting cubic circle graphs

We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition. We also deduce that up to isomorphism, K_4 and K_{3,3} are the only 3-connected, 3-regular circle graphs.

Characterizing graph classes using twin vertices of regular induced subgraphs

Characterizing graph classes using twin vertices of regular induced subgraphs

Multiple Petersen Subdivisions in Permutation Graphs

A permutation graph is a cubic graph admitting a 1-factor $M$ whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if $e$ is an edge of $M$ such that every 4-cycle containing an edge of $M$ contains $e$, then $e$ is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of $M$ is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor.

A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

The number of convergent graphs under the biclique operator with no twin vertices is finite

The biclique graph of G, KB(G), is the intersection graph of the bicliques of G. Given a graph G, the iterated biclique graph of G, KBk(G), is the graph defined iteratively as follows: KBk+1(G)=KB(KBk(G)). Say that a graph G diverges (resp. converges) under the operator KB whenever limk→∞V(KBk(G))=∞ (resp. limk→∞KBk(G)=KBm(G) for some m). Each of these behaviours were recently characterized. These characterizations lead to a O(n4) time algorithm for deciding the divergence or convergence of a graph. In this work we prove that any graph with at least 7 bicliques diverges under the biclique operator. Furthermore, we prove that graphs with no twin vertices that are not divergent have at most 12 vertices, which leads to a linear time algorithm to decide if a graph converges or diverges under the biclique operator.