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Author

A. A. Makhnev

Other affiliations: Ural Federal University
Bio: A. A. Makhnev is an academic researcher from Russian Academy of Sciences. The author has contributed to research in topics: Strongly regular graph & Distance-regular graph. The author has an hindex of 6, co-authored 80 publications receiving 162 citations. Previous affiliations of A. A. Makhnev include Ural Federal University.


Papers
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Journal ArticleDOI
TL;DR: In this paper, it was proved that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array: $$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2,
Abstract: A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ1 equal to a3. For a Shilla graph, let us put a = a3 and b = k/a. It is proved in this paper that a Shilla graph with b2 = c2 and noninteger eigenvalues has the following intersection array: $$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$ If Γ is a Q-polynomial Shilla graph with b2 = c2 and b = 2r, then the graph Γ has intersection array $$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$ and, for any vertex u in Γ, the subgraph Γ3(u) is an antipodal distance-regular graph with intersection array $$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$ The Shilla graphs with b2 = c2 and b = 4 are also classified in the paper.

11 citations

Journal ArticleDOI
TL;DR: In this article, the authors proposed a classification for the graphs based on the description of twice transitive permutation groups for affine bipartite clique coverings of edge-symmetric distance-regular covering of cliques.
Abstract: Let Γ be an edge-symmetric distance-regular covering of a clique. Then the group G = Aut(Γ) acts twice transitively on the set Σ of antipodal classes. We propose a classification for the graphs based on the description of twice transitive permutation groups. This program is realized for a1 = c2. In this article we classify graphs in the case when the action of G on Σ is affine.

8 citations

Journal ArticleDOI
05 Apr 2014
TL;DR: In this paper, the authors classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors, where {a, b} is an edge.
Abstract: We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors.

7 citations


Cited by
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Book ChapterDOI
01 Jan 2007

1,089 citations

Journal ArticleDOI
TL;DR: Several vertex distance-based indices of hexagonal jagged-rectangle from the mathematical point of view are determined and the Wiener related indices of distance-regular structure are considered.
Abstract: In theoretical medicine, topological indices are defined to test the medicine and pharmacy characteristics, such as melting point, boiling point, toxicity and other biological activities. As basic molecular structures, hexagonal jagged-rectangle and distance-regular structure are widely appeared in medicine, pharmacy and biology engineering. In this paper, we study the chemical properties of hexagonal jagged-rectangle from the mathematical point of view. Several vertex distance-based indices are determined. Furthermore, the Wiener related indices of distance-regular structure are also considered.

98 citations

Posted Content
TL;DR: An introduction to distance-regular graphs is presented for the reader who is unfamiliar with the subject, and an overview of some developments in the area of distance- regular graphs since the monograph 'BCN' was written.
Abstract: This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

46 citations

Journal ArticleDOI
TL;DR: A priori accuracy estimates for low-rank matrix approximations that use just a small number of the rows and columns are proposed, which are more accurate than previously known ones.

44 citations