ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.

Presentations for vertex-transitive graphs

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph.

Method of using the transitive graph of a Markovian process as part of ranking of heterogeneous items

Hierarchy analysis developed by Thomas Saaty is a closed logical structure that uses simple and well-substantiated rules that allow solving multicriterial problems that include both quantitative, and qualitative factors, whereby the quantitative factors can differ in terms of their dimensionality. The method is based on problem decomposition and its representation as a hierarchical arrangement, which allows including into such hierarchy all the knowledge the decision-maker has regarding the problem at hand and subsequent processing of decision-makers’ judgements. As the result, the relative degree of the interaction between the elements of such hierarchy can be identified and later quantified. Hierarchy analysis includes the procedure of multiple judgement synthesis, criteria priority definition and rating of the compared alternatives. The method’s significant limitation consists in the requirement of coherence of pairwise comparison matrices for correct definition of the weights of compared alternatives. The Aim of the paper is to examine a non-conventional method of solving the problem of alternative ratings estimation based on their pairwise comparisons that arises in the process of expert preference analysis in various fields of research. Approaches are discussed to the generation of pairwise comparison matrices taking into consideration the problem of coherence of such matrices and expert competence estimation. Method. The methods of hierarchy analysis, models and methods of the Markovian process theory were used. Result. The paper suggested a method of using the transitive graph of a Markovian process as part of expert ranking of items of a certain parent entity subject to the competence and qualification of the experts involved in the pairwise comparison. It is proposed to use steady-state probabilities of a Markovian process as the correlation of priorities (weights) of the compared items. The paper sets forth an algorithm for constructing the final scale of comparison taking into consideration the experts’ level of competence. Conclusion. The decision procedures, in which the experts are expected to choose the best alternatives out of the allowable set, are quite frequently used in a variety of fields for the purpose of estimation and objective priority definition, etc. The described method can be applied not only for comparing items, but also for solving more complicated problems of expert group estimation, i.e., planning and management, prediction, etc. The use of the method contributes to the objectivity of analysis, when comparing alternatives, taking into consideration various aspects of their consequences, as well as the decision-maker’s attitude to such consequences. The suggested model-based approach allows the decision-maker identifying and adjusting his/her preferences and, consequently, choosing the decisions according to such preferences, avoiding logical errors in long and complex reasoning chains. This approach can be used in group decision-making, description of the procedures that compensate a specific expert’s insufficient knowledge by using information provided by the other experts.

CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T} _d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound. It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

Two-Distance-Primitive Graphs

A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let $\Gamma$ be such a graph. This paper shows that either $\Gamma$ is a cyclic graph, or $\Gamma$ is a complete bipartite graph, or $\Gamma$ has girth at most $4$ and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs satisfying that the vertex stabilizer acts $2$-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is locally cyclic.

Cayley Graphs Without a Bounded Eigenbasis

Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

Analysis of lackadaisical quantum walks

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any regular locally arc-transitive graph.

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and $$p > p_c(G)$$ p > p c ( G ) then there exists a positive constant $$c_p$$ c p such that $$\begin{aligned} \mathbf {P}_p(n \le |K| < \infty ) \le e^{-c_p n} \end{aligned}$$ P p ( n ≤ | K | < ∞ ) ≤ e - c p n for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We deduce the following two corollaries: Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999). For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

No Percolation at Criticality on Certain Groups of Intermediate Growth

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp \left [-\Omega (n^\gamma )\right ]$ for some $\gamma&gt;1/2$. The result is new in the case that the graph is of intermediate volume growth.

A survey on Hamiltonicity in Cayley graphs and digraphs on different groups

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.