On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k ? 1) + ... + k(k ? 1)^d?1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k ² + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).     

Anisotropy of the Complex Permittivity of the Kagome-Staircase Compounds Co<Subscript>3</Subscript>V<Subscript>2</Subscript>O<Subscript>8</Subscript> and Ni<Subscript>3</Subscript>V<Subscript>2</Subscript>O<Subscript>8</Subscript>: Experiment and Ab Initio Calculations

The anisotropy of the components of the complex permittivity of vanadate Co₃V₂O₈ and Co₃V₂O₈ single crystals in the paramagnetic phase are studied by optical ellipsometry in the spectral region 0.5–5.0 eV. Our experimental results support the weak anisotropy of the optical response detected earlier for axes a and c. The optical properties are also investigated along axis b. The properties of both compounds are compared. The optical spectra of both compounds along axis b are shifted toward low energies as compared to axes a and c. The maximum of the main interband absorption band of Co₃V₂O₈ is shifted toward low energies by 0.25–0.3 eV as compared to Co₃V₂O₈. The electronic structure parameters of both compounds are determined. Optical function spectra are analyzed using the results of ab initio band calculations.     

Evolution of the optical properties and the electrical resistivity of CaMnO3 during the substitution of Mo for Mn ions

The reflectance, the optical density, and the optical conductivity spectra of CaMn_{1 ? x} Mo _x O₃ (x ≤ 0.07) polycrystals are studied over a wide spectral region. The substitution of Mo⁶⁺ ions for Mn⁴⁺ ions leads to a strong decrease in the first high-energy phonon band, which is related to vibrations in the MnO₆ octahedron. The appearance of band-type charge carriers is detected in CaMn_{1 ? x} Mo _x O₃ with x ≤ 0.04, and carrier localization is observed in CaMn_0.93Mo_0.07O₃. This localization increases at T < 160 K, i.e., during the transition from the paramagnetic to the antiferromagnetic state with orbital ordering and a monoclinic structure. The optical conductivity spectrum of CaMn_0.93Mo_0.07O₃ exhibits a strong shift of fundamental absorption bands, which is associated with a change in the band structure because of a decrease in the splitting of the 3d levels of Mn in the crystal field.     

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

Mathematical Notes - A. A. Makhnev and M. S. Nirova found the intersection arrays of distance regular graphs with $$\lambda=2$$ and at most 4096 vertices. For graphs of diameter $$4$$ , of most...     

Antipodal Krein Graphs and Distance-Regular Graphs Close to Them

Doklady Mathematics - An antipodal nonbipartite distance-regular graph Γ of diameter 3 has an intersection array $$\{ k,(r - 1){{c}_{2}},\;1;\;1,\;{{c}_{2}},\;k\} $$ ( $${{c}_{2}} &lt; k -...     

On Strongly Regular Graphs with k=2μ and Their Extensions

We obtain a convenient expression for the parameters of a strongly regular graph with k=2 in terms of the nonprincipal eigenvalues x and –y. It turns out in particular that such graphs are pseudogeometric for pG _x(2x,y–1). We prove that a strongly regular graph with parameters (35,16,6,8) is a quotient of the Johnson graph (8,4). We also find the parameters of strongly regular graphs in which the neighborhoods of vertices are pseudogeometric graphs for pG _x(2x,t),x3. In consequence, we establish that a connected graph in which the neighborhoods of vertices are pseudogeometric graphs for pG ₃(6,2) is isomorphic to the Taylor graph on 72 vertices or to the alternating form graph Alt(4,2) with parameters (64,35,18,20).