A general construction of strictly Neumaier graphs and related switching

In this paper we propose a construction of Neumaier graphs with nexus 1, which generalises two known constructions. We then discuss small strictly Neumaier graphs obtained from the general construction and give a geometric description for some of them. Finally, we apply a variation of the Godsil-McKay switching to the general construction

Equitable partitions of Latin-square graphs

Funding: R.A. Bailey and Peter J. Cameron are grateful to Shanghai Jiao Tong University for funding, from the National Science Foundation of China (11671258) and STCSM (17690740800), a research visit where part of this study was done. Alexander L. Gavrilyuk was supported by BK21plus Center for Math Research and Education at Pusan National University, Republic of Korea. Sergey V. Goryainov was supported by the National Science Foundation of China, STCSM (17690740800) and RFBR (17‐51‐560008).We study equitable partitions of Latin-square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue -3.PostprintPeer reviewe

Deza graphs with parameters (n,k,k−1,a)(n,k,k-1,a) and β=1β=1

A Deza graph with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices in which any two vertices have $a$ or $b$ ($a\leq b$) common neighbours. A Deza graph is strictly Deza if it has diameter $2$, and is not strongly regular. In an earlier paper, the two last authors et el. characterized the strictly Deza graphs with $b=k-1$ and $\beta > 1$, where $\beta$ is the number of vertices with $b$ common neighbours with a given vertex. Here we deal with the case $\beta=1$, thus we complete the characterization of strictly Deza graphs with $b=k-1$. It follows that all Deza graphs with $b=k-1$ and $\beta=1$ can be made from special strongly regular graphs, and we present several examples of such strongly regular graphs. A divisible design graph is a special Deza graph, and a Deza graph with $\beta=1$ is a divisible design graph. The present characterization reveals an error in a paper on divisible design graphs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata

Equitable partitions of Latin-square graphs

We study equitable partitions of Latin-square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue -3

Ultrahigh-Temperature Sphalerite from Zn-Cd-Se-Rich Combustion Metamorphic Marbles, Daba Complex, Central Jordan: Paragenesis, Chemistry, and Structure

Minerals of the Zn-Cd-S-Se system that formed by moderately reduced ~800–850 °C combustion metamorphic (CM) alteration of marly sediments were found in marbles from central Jordan. Their precursor sediments contain Se- and Ni-enriched authigenic pyrite and ZnS modifications with high Cd enrichment (up to ~10 wt%) and elevated concentrations of Cu, Sb, Ag, Mo, and Pb. The marbles are composed of calcite, carbonate-fluorapatite, spurrite, and brownmillerite and characterized by high P, Zn, Cd, U, and elevated Se, Ni, V, and Mo contents. Main accessories are either Zn-bearing oxides or sphalerite, greenockite, and Ca-Fe-Ni-Cu-O-S-Se oxychalcogenides. CM alteration lead to compositional homogenization of metamorphic sphalerite, for which trace-element suites become less diverse than in the authigenic ZnS. The CM sphalerites contain up to ~14 wt% Cd and ~6.7 wt% Se but are poor in Fe (means 1.4–2.2 wt%), and bear 100–250 ppm Co, Ni, and Hg. Sphalerite (Zn,Cd,Fe)(S,O,Se)cub is a homogeneous solid solution with a unit cell smaller than in ZnScub as a result of S2− → O2− substitution (a = 5.40852(12) Å, V = 158.211(6) Å3). The amount of lattice-bound oxygen in the CM sphalerite is within the range for synthetic ZnS1−xOx crystals (0 < x ≤ 0.05) growing at 900 °C