Automorphisms Of Graph Research Articles

Eccentric topological properties of a graph associated to a finite dimensional vector space

Abstract A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex-induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of standard tools of inverse semigroup theory, namely Green’s relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.

The Largest Moore Graph and a Distance-Regular Graph with Intersection Array {55, 54, 2; 1, 1, 54}

We point out possible automorphisms of a distance-regular graph Γ with intersection array {55, 54, 2; 1, 1, 54} and spectrum 551, 71617,−1110,−81408.

Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

The Graovac–Pisanski (GP) index of a graph is a modified version of the Wiener index based on the distance between each vertex x and its image α(x), where α is an automorphism of graph. The aim of this paper is to compute the automorphism group of some classes of cubic polyhedral graphs and then we determine their Wiener index. In addition, we investigate the GP-index of these classes of graphs.

The Graphical Conditions for Controllability of Multiagent Systems Under Equitable Partition.

In this article, by analyzing the eigenvalues and eigenvectors of Laplacian L , we investigate the controllability of multiagent systems under equitable partitions. Two classes of nontrivial cells are defined according to the different numbers of links between them, which are completely connected nontrivial cells (CCNCs) and incompletely connected nontrivial cells. For the system with CCNCs, a necessary condition for controllability is found to be choosing leaders from each nontrivial cell, the number of which should be one less than the cardinality of the cell. It is shown that the controllability is affected by three factors: 1) the number of the links between nontrivial cells; 2) the rank of the connection matrix; and 3) the odevity of the capacity of the nontrivial cells. In the case of nontrivial cells under the equitable partition, there are automorphisms of interconnection graph G , which induce the eigenvectors of L with zero entries. For the system with automorphisms, by taking advantage of the property of eigenvectors associated with L , we propose several graphical necessary conditions for controllability. In addition, by the PBH rank criterion, the controllable subspaces of the system with different classes of nontrivial cells are compared. Finally, a necessary and sufficient condition for controllability under minimum inputs is given.

A Graph Automorphic Approach for Placement and Sizing of Charging Stations in EV Network Considering Traffic

This paper proposes a novel graph-based approach with automorphic grouping for the modelling, synthesis, and analysis of electric vehicle (EV) networks with charging stations (CSs) that considers the impacts of traffic. The EV charge demands are modeled by a graph where nodes are positioned at potential locations for CSs, and edges represent traffic flow between the nodes. A synchronization protocol is assumed for the network where the system states correspond to the waiting time at each node. These models are then utilized for the placement and sizing of CSs in order to limit vehicle waiting times at all stations below a desirable threshold level. The main idea is to reformulate the CS placement and sizing problems in a control framework. Moreover, a strategy for the deployment of portable charging stations (PCSs) in selected areas is introduced to further improve the quality of solutions by reducing the overshooting of waiting times during peak traffic hours. Further, the inherent symmetry of the graph, described by graph automorphisms, are leveraged to investigate the number and positions of CSs. Detailed simulations are performed for the EV network of Perth Metropolitan in Western Australia to verify the effectiveness of the proposed approach.

Computing autotopism groups of partial Latin rectangles: A pilot study

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by $2 \times 2$ submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick-to-compute entry invariant was effective at reducing the need for computation, (c) with an almost-full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performance for (full) Latin squares is significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of Latin squares with large (possibly transitive) autotopism groups.

Arc-transitive groups of automorphisms of antipodal distance-regular graphs of diameter 3 in affine case

Arc-transitive groups of automorphisms of antipodal distance-regular graphs of diameter 3 in affine case

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-torus

This paper is devoted to the study of special subgroups of the automorphism groups of Kronrod-Reeb graphs of a Morse functions on $2$-torus $T^2$ which arise from the action of diffeomorphisms preserving a given Morse function on $T^2$. In this paper we give a full description of such classes of groups.

Twisted Lawrence-Krammer representations

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected.Here, we investigate an analogue of these representations — introduced by Digne in the spherical cases — for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group.Under the same assumptions on the parameters as in the small type cases, we first show that these so-called “twisted Lawrence-Krammer representations” are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one.

Characterizing cospectral vertices via isospectral reduction

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices.

ON CONNECTED TETRAVALENT NORMAL EDGE-TRANSITIVE CAYLEY GRAPHS OF NON-ABELIAN GROUPS OF ORDER 5p^2

Our aim in this paper is to investigate graph automorphism and group automorphism determining all connected tetravalent normal edge transitive Cayley graphs on non-Abelian groups of order 5p2 with respect to tetravalent sets and same-order elements, where p is a prime number and its Sylow p-subgroup is cyclic.

Deformations of smooth functions on 2-torus

Let $f$ be a Morse function on a smooth compact surface $M$ and $\mathcal{S}'(f)$ be the group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the Kronrod-Reeb graph of $f$ induced by elements from $\mathcal{S}'(f)$, and $\Delta'$ be the subgroup of $\mathcal{S}'(f)$ consisting of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $\pi_0\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. The groups $\pi_0\Delta'(f)$ and $G(f)$ encode ``combinatorially trivial'' and ``combinatorially nontrivial'' counterparts of $\pi_0\mathcal{S}'(f)$ respectively. In the paper we compute groups $\pi_0\mathcal{S}'(f)$, $G(f)$, and $\pi_0\Delta'(f)$ for Morse functions on $2$-torus $T^2$.

Geometric realizations of cyclic actions on surfaces

Let [Formula: see text] denote the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text], and let [Formula: see text] be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on [Formula: see text] that realizes [Formula: see text] as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of [Formula: see text]. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation [Formula: see text].

Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces

Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of difeomorphisms of $M$ isotopic to the identity. Denote by $\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms "preserving" $f$, i.e. the stabilizer of $f$ with respect to the right action of $\mathcal{D}_{\mathrm{id}}(M)$ on the space $\mathcal{C}^{\infty}(M,\mathbb{R})$ of smooth functions on $M$. Let also $\mathbf{G}(f)$ be the group of automorphisms of the Kronrod-Reeb graph of $f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is an important ingredient in determining the homotopy type of the orbit of $f$ with respect to the above action of $\mathcal{D}_{\mathrm{id}}(M)$ and it is trivial if $f$ is "generic", i.e. has at most one critical point at each level set $f^{-1}(c)$, $c\in\mathbb{R}$. For the case when $M$ is distinct from $2$-sphere and $2$-torus we present a precise description of the family $\mathbf{G}(M,\mathbb{R})$ of isomorphism classes of groups $\mathbf{G}(f)$, where $f$ runs over all Morse functions on $M$, and of its subfamily $\mathbf{G}^{smp}(M,\mathbb{R}) \subset \mathbf{G}(M,\mathbb{R})$ consisting of groups corresponding to simple Morse functions, i.e. functions having at most one critical point at each connected component of each level set. In fact, $\mathbf{G}(M,\mathbb{R})$, (resp. $\mathbf{G}^{smp}(M,\mathbb{R})$), coincides with the minimal family of isomorphism classes of groups containing the trivial group and closed with respect to direct products and also with respect to wreath products "from the top" with arbitrary finite cyclic groups, (resp. with group $\mathbb{Z}_2$ only).

Strict monotonicity of percolation thresholds under covering maps

We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_{c}$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_{c}(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_{c}(\mathcal{G})<p_{c}(\mathcal{H})$. We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the uniqueness parameter $p_{u}$, under an additional assumption of boundedness of the fibres. The proof makes use of a coupling built by lifting the exploration of the cluster, and an exploratory counterpart of Aizenman–Grimmett’s essential enhancements.

Automorphisms of distance-regular graph with intersection array $\{24,18,9;1,1,16\}$

Automorphisms of distance-regular graph with intersection array $\{24,18,9;1,1,16\}$

Automorphisms of Grassmann graphs over a residue class ring

Let Zps be the residue class ring of integers modulo ps, where p is a prime number and s is a positive integer. The Grassmann graph over Zps, denoted by G(n,m,ps), has the vertex set all m-subspaces of Zpsn (n>m≥1), and two vertices are adjacent if and only if their intersection is of dimension m−1. We characterize the automorphisms of G(n,m,ps) as follows. Let n≥2m≥4 and let φ∈Aut(G(n,m,ps)). Then either φ(X)=XU for all X∈V(G(n,m,ps)), or n=2m and φ(X)=(XU)⊥ for all X∈V(G(2m,m,ps)), where U is a fixed invertible matrix and (XU)⊥ is the dual subspace of XU. This result also extends Chow’s theorem for the geometry of Grassmann space.

Automorphisms of small graphs with intersection array $\{nm-1,nm-n+m-1,n-m+1;1,1,nm-n+m-1\}$

Automorphisms of small graphs with intersection array $\{nm-1,nm-n+m-1,n-m+1;1,1,nm-n+m-1\}$