A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $a_0$-regular and for every vertex $v$ of $X$, the sphere radius $1$ around an $(a_1,\dots,a_{n-1})$-regular graph. Such a said highly regular (HR) level $n$ $a_{n-1}\neq 0$. Chapman, Linial Peled studied HR-graphs 2 provided several methods construct families graphs which are expanders globally locally. They ask whether ...

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K3,3 and the s-regular cyclic or elementary abelian coverings of the complete graph K4 for each s 1 are classified when the fibrepreserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regu...

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of S+(U3), a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of orde...

A graph G is called locally s-regular if the neighbourhood of each vertex of G induces a subgraph of G which is regular of degree s. We study graphs which are locally s-regular and simultaneously regular of degree r.

For 3 ≤ k ≤ 20 with k 6= 4, 8, 12, all the smallest currently known k–regular graphs of girth 5 have the same orders as the girth 5 graphs obtained by the following construction: take a (not necessarily Desarguesian) elliptic semiplane S of order n− 1 where n = k − r for some r ≥ 1; the Levi graph Γ (S) of S is an n–regular graph of girth 6; parallel classes of S induce co–cliques in Γ (S), som...

In a bi-racial country, it is desired that the numbers of government ministers from the two races should differ by at most one for fairness. Moreover, the numbers of pairs of ministries which interact directly with each other and both headed by ministers of one race should differ by at most one from that of the other race. One can naturally model the above social phenomenon by a graph labeling ...