相关论文: Neumaier graphs from cyclotomy with small coherent…
We characterise graphs that have three distinct eigenvalues and coherent ranks 8 and 9, linking the former to certain symmetric 2-designs and the latter to specific quasi-symmetric 2-designs. This characterisation leads to the discovery of…
We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show…
We give inductive constructions of independent graphs that contain implied nonedges but do not contain any non-trivial rigid subgraphs, or \emph{nucleations}: some of the constructions and proofs apply to 3-dimensional abstract rigidity…
In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using…
Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W.~Kantor's non-classical $\mathrm{GQ}(5^2,5)$, are stumbling stones for existing…
In this paper, we introduce a new family of graphs, $\Gamma(n,a)$. We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of $\Gamma(n,a)$.
We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on…
We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled…
In this paper we introduce a construction of directed strongly regular graphs from smaller ones using equitable partitions. Each equitable partition of a single DSRG satisfying several conditions leads to an infinite family of directed…
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…
We exhibit infinitely many examples of edge-regular graphs that have regular cliques and that are not strongly regular. This answers a question of Neumaier from 1981.
Euler graphs are characterized by the simple criterion that degree of each node is even. By restricting on the cycle types yet additional intrinsic properties of Euler graphs are unveiled. For example, regularity higher than degree two is…
A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes known constructions for…
We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a non-trivial partition, and symmetric and alternating groups.
We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic…
A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as…
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions…
In this paper we define two infinite families of graphs called C-$\delta$ graphs and $\delta$- graph and prove that $\delta$-graphs satisfy $\delta$ conjecture. Also we introduce a family of C-$\delta$ graphs from which we can identify…
We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[\zeta_m]/A$, with connection sets $\{\pm (\zeta_m^i + A): 0 \le i \le m-1\}$ and $\{\pm (\zeta_m^i + A): 0 \le…