A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

On intersections of conjugate subgroups

We define a new invariant of a conjugacy class of subgroups which we call the breadth and prove that a quasiconvex subgroup of a negatively curved group has finite breadth in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex subgroup of a negatively curved group with its conjugate. Using that algorithm we construct algorithms for computing the breadth, the width, and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group. We also explicitly compute a quasiconvexity constant of the intersection of two quasiconvex subgroups and give examples demonstrating that height, width, and breadth are different invariants of a subgroup.

Pentavalent Symmetric Graphs of Order Twice a Prime Square

A classification of pentavalent symmetric graphs of order twice a prime square is given. It is proved that such a graph is a coset graph of ℤ3. A 6 (non-split extension), or a bi-coset graph of an extra-special group of order 125, or the standard double cover of a specific abelian Cayley digraph of order a prime square.

PENTAVALENT SYMMETRIC GRAPHS OF ORDER

A complete classification is given of pentavalent symmetric graphs of order$30p$, where$p\ge 5$is a prime. It is proved that such a graph${\Gamma }$exists if and only if$p=13$and, up to isomorphism, there is only one such graph. Furthermore,${\Gamma }$is isomorphic to$\mathcal{C}_{390}$, a coset graph of PSL(2, 25) with${\sf Aut}\, {\Gamma }=\mbox{PSL(2, 25)}$, and${\Gamma }$is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order.