Automorphisms Of Graph Research Articles

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 $\{nm-1,nm-n+m-1,n-m+1;1,1,nm-n+m-1\}$

New Exact and Heuristic Algorithms for Graph Automorphism Group and Graph Isomorphism

We describe five new algorithms, named Vsep. Four of them are for the graph automorphism group and the fifth one is for finding an isomorphism between two graphs. All nonequivalent terminal nodes-discrete partitions of the search tree are stored. This is the main difference of the exact version with the known algorithms for graph automorphism group. A new strategy is used in the exact algorithm: if during its execution the computed stabilizer orbits and order get wrong values, then the algorithm continues from a new starting point losing some of the results determined so far. The new starting point is such that the results are correct. The proposed algorithms have been tested on well-known benchmark graphs and compared with three of the most competitive known tools. The results show that for some graph families the exact Vsep algorithm outperforms these algorithms, and vice-versa for some of the others. None of the tested algorithms outperform others in all cases. The heuristic versions use reduced search trees. They are almost exact and are faster than the exact one with very rare exceptions. They are applied mainly for regular graphs. The heuristic algorithms are a new choice for the user. The experiments show that the running times of Vsep algorithms have a slight dependence on vertex labeling.

K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F

We compute the K-automorphism group of AK=K⊗FAf of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent part and the reductive part Hˆ. Automorphisms in Hˆ are computed via their restriction to semi-simple part of AK. There is a strong relationship between Hˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of Hˆ, namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

Automorphisms of the total graph over upper triangular matrices

Let [Formula: see text] be a finite field, [Formula: see text] the ring of all [Formula: see text] upper triangular matrices over [Formula: see text], [Formula: see text] the set of all zero-divisors of [Formula: see text], i.e. [Formula: see text] consists of all [Formula: see text] upper triangular singular matrices over [Formula: see text]. The total graph of [Formula: see text], denoted by [Formula: see text], is a graph with all elements of [Formula: see text] as vertices, and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine all automorphisms of the total graph [Formula: see text] of [Formula: see text].

Compressed zero-divisor graphs of matrix rings over finite fields

ABSTRACT We extend the notion of the compressed zero-divisor graph to noncommutative rings in a way that still induces a product preserving functor from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of , the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when . For we prove that every graph automorphism of is induced by a ring automorphism of . We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if , then . In particular, this holds if with .

The Hosoya Entropy of Graphs Revisited

In this paper we extend earlier results on Hosoya entropy (H-entropy) of graphs, and establish connections between H-entropy and automorphisms of graphs. In particular, we determine the H-entropy of graphs whose automorphism group has exactly two orbits, and characterize some classes of graphs with zero H-entropy.

Automorphisms of a Distance-Regular Graph with Intersection Array {176, 135, 32, 1; 1, 16, 135, 176}

A distance-regular graph Γ with intersection array {176, 135, 32, 1; 1, 16, 135, 176} is an AT4-graph. Its antipodal quotient \(\overline {\rm{\Gamma }} \) is a strongly regular graph with parameters (672, 176, 40, 48). In both graphs the neighborhoods of vertices are strongly regular with parameters (176, 40, 12, 8). We study the automorphisms of these graphs. In particular, the graph Γ is not arc-transitive. If G = Aut (Γ) contains an element of order 11, acts transitively on the vertex set of Γ, and S(G) fixes each antipodal class, then the full preimage of the group (G/S(G))′ is an extension of a group of order 3 by M22 or U6 (2). We describe automorphism groups of strongly regular graphs with parameters (176, 40, 12, 8) and (672, 176, 40, 48) in the vertex-symmetric case.

Action of a Graph Automorphism on the Space of Flows

Action of a Graph Automorphism on the Space of Flows

On automorphisms of a distance-regular graph with intersection array $\{44,30,5;1,3,40\}$

On automorphisms of a distance-regular graph with intersection array $\{44,30,5;1,3,40\}$

On automorphisms of a distance-regular graph with intersection array $\{39,36,22;1,2,18\}$

On automorphisms of a distance-regular graph with intersection array $\{39,36,22;1,2,18\}$

Lifting a prescribed group of automorphisms of graphs

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Let Γ \Gamma be a finite graph and let A u t ( Γ ) \mathrm {Aut}(\Gamma ) be the automorphism group of Γ \Gamma . It is well known that we can always find a finite graph Γ ~ \tilde {\Gamma } and a regular covering projection ℘ : Γ ~ → Γ \wp \colon \tilde {\Gamma } \to \Gamma such that A u t ( Γ ) \mathrm {Aut}(\Gamma ) lifts along ℘ \wp . However, for constructing peculiar examples and in applications it is often important, given a subgroup G G of A u t ( Γ ) \mathrm {Aut}(\Gamma ) , to find ℘ \wp along which G G lifts but no further automorphism of Γ \Gamma does, or even that A u t ( Γ ~ ) \mathrm {Aut}(\tilde {\Gamma }) is the lift of G G . In this paper, we address these problems.

On automorphisms of graphs and Riemann surfaces acting with fixed points

Let X be a finite connected graph, possibly with multiple edges. We provide each edge of the graph by two possible orientations. An automorphism group of a graph acts harmonically if it acts freely on the set of directed edges of the graph. Following M. Baker and S. Norine define a genus g of the graph X to be the rank of the first homology group. A finite group acting harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g. In the present paper, we give a sharp upper bound for the size of cyclic group acting harmonically on a graph of genus $$g\ge 2$$ with a given number of fixed points. Similar results, for closed orientable surfaces, were obtained earlier by T. Szemberg, I. Farkas and H. M. Kra.

Automorphisms of distance regular graph with intersection array $\{30,27,24;1,2,10\}$

Automorphisms of distance regular graph with intersection array $\{30,27,24;1,2,10\}$

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

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$.
 In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere. 
 The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs

A non-trivial automorphism g of a graph Γ is called semiregular if the only power gi fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power gi fixing another vertex is the identity mapping. In this paper, we prove that K4, the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and K5 is the only connected tetravalent 2-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent G-arc-transitive graph, where G is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order.

Lipschitz-free Spaces on Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

Formula omitted]-regular families of graph automorphisms

An r-regular family F of permutations on a set V contains, for each pair of vertices u,v∈V, exactly r permutations φ mapping u to v. Earlier, 1-regular families of graph automorphisms were used by Gauyacq to define the quasi-Cayley graphs, a class of vertex-transitive graphs that properly contains the class of Cayley graphs, sharing many of their characteristics, and is properly contained in the class of vertex-transitive graphs. We introduce r-regular families to measure how far a vertex-transitive graph is from being quasi-Cayley. As any automorphism group of a graph Γ=(V,E) acting transitively on V with vertex-stabilizers of order r forms an r-regular family on V, every vertex-transitive graph admits an r-regular family of automorphisms for some r≥1. In general, the smallest r for which such a family exists (which we call the quasi-Cayley deficiency of the graph) might be smaller than the order of the vertex-stabilizer of a smallest vertex-transitive automorphism group of the graph (which we call the Cayley deficiency). We investigate the relations between these two parameters for the class of merged Johnson graphs. We prove the existence of Johnson graphs with arbitrarily large quasi-Cayley deficiency, as well as Johnson graphs for which the difference between their Cayley deficiency and their quasi-Cayley deficiency is arbitrarily large.

Action of a Graph Automorphism on the Space of Flows

Action of a Graph Automorphism on the Space of Flows

Inverse Problems in Graph Theory: Nets

Let $$\varGamma $$ be a distance-regular graph of diameter 3 with strong regular graph $$\varGamma _3$$ . The determination of the parameters $$\varGamma _3$$ over the intersection array of the graph $$\varGamma $$ is a direct problem. Finding an intersection array of the graph $$\varGamma $$ with respect to the parameters $$\varGamma _3$$ is an inverse problem. Previously, inverse problems were solved for $$\varGamma _3$$ by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph $$\varGamma $$ of diameter 3, for which the graph $${\bar{\varGamma }}_3$$ is a pseudo-geometric graph of the net $$PG_{m}(n, m)$$ . New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array $$\{20,16,5; 1,1,16 \}$$ .

On Automorphisms of a Distance-Regular Graph with Intersection Array {69, 56, 10; 1, 14, 60}

Let Γ be a distance-regular graph of diameter 3 with eigenvalues θ0 > θ1 > θ2 > θ3. If θ2 = −1, then the graph Γ3 is strongly regular and the complementary graph \({\bar \Gamma _3}\) is pseudogeometric for pGc3(k, b1/c2). If Γ3 does not contain triangles and the number of its vertices v is less than 800, then Γ has intersection array {69, 56, 10; 1, 14, 60}. In this case Γ3 is a graph with parameters (392, 46, 0, 6) and \({\bar \Gamma _2}\) is a strongly regular graph with parameters (392, 115, 18, 40). Note that the neighborhood of any vertex in a graph with parameters (392, 115, 18, 40) is a strongly regular graph with parameters (115, 18, 1, 3) and its existence is unknown. In this paper, we find possible automorphisms of these strongly regular graphs and automorphisms of a hypothetical distance-regular graph with intersection array {69, 56, 10; 1, 14, 60}. In particular, it is proved that the latter graph is not ar-ctransitive.