A reflection of a regular map on a Riemann surface fixes some simple closed curves, which are called \({mirrors}\). Each mirror passes through some of the geometric points (vertices, face-centers and edge-centers) of the map such that these points form a periodic sequence which we call the \({pattern}\) of the mirror. For every mirror there exist two particular conformal automorphisms of the map that fix the mirror setwise and rotate it in opposite directions. We call these automorphisms the \({rotary\; automorphisms}\) of the mirror. In this paper, we first introduce the notion of pattern and then describe the patterns of mirrors on surfaces. We also determine the rotary automorphisms of mirrors. Finally, we give some necessary conditions under which all reflections of a regular map are conjugate.

We prove the non-existence of maximal partial spreads of size \(76\) in \(\text{PG}(3,9)\). Relying on the classification of the minimal blocking sets of size 15 in \(\text{PG}(2,9)\) \([22]\), we show that there are only two possibilities for the set of holes of such a maximal partial spread. The weight argument of Blokhuis and Metsch \([3]\) then shows that these sets cannot be the set of holes of a maximal partial spread of size \(76\). In \([17]\), the non-existence of maximal partial spreads of size \(75\) in \(\text{PG}(3,9)\) is proven. This altogether proves that the largest maximal partial spreads, different from a spread, in \(\text{PG}(3,q = 9)\) have size \(q^2 – q + 2 = 74\).

A weakly connected dominating set \(W\) of a graph \(G\) is a dominating set such that the subgraph consisting of \(V(G)\) and all edges incident on vertices in \(W\) is connected. In this paper, we generalize it to \([r, R]\)-dominating set which means a distance \(r\)-dominating set that can be connected by adding paths with length within \(R\). We present an algorithm for finding \([r, R]\)-dominating set with performance ratio not exceeding \(ln \Delta_r + \lceil \frac{2r+1}{R}\rceil – 1\), where \(\Delta_r\) is the maximum number of vertices that are at distance at most \(r\) from a vertex in the graph. The bound for size of minimum \([r, R]\)-dominating set is also obtained.

For \(n \in \mathbb{N}\), let \(a_n\) count the number of ternary strings of length \(n\) that contain no consecutive \(1\)s. We find that \(a_n = \left(\frac{1}{2}+\frac{\sqrt{3}}{3}\right)\left(1 + \sqrt{3}\right)^n – \left(\frac{1}{2}-\frac{\sqrt{3}}{3}\right)\left(1 – \sqrt{3}\right)^n\). For a given \(n \geq 0\), we then determine the following for these \(a_n\) ternary strings:
(1)the number of \(0’\)s, \(1’\)s, and \(2’\)s;(2)the number of runs;(3) the number of rises, levels, and descents; and
(4)the sum obtained when these strings are considered as base \(3\) integers.

Following this, we consider the special case for those ternary strings (among the \(a_n\) strings we first considered) that are palindromes, and determine formulas comparable to those in (1) – (4) above for this special case.

Topological indices of nanotubes are numerical descriptors that are derived from the graph of chemical compounds. Such indices, based on the distances in the graph, are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Szeged index is obtained as a bond additive quantity, where bond contributions are given as the product of the number of atoms closer to each of the two end points of each bond. In this paper, we find an exact expression for the Szeged index of an armchair polyhex nanotube \((TUAC_6{[p,k]}\)).

It is widely recognized that certain graph-theoretic extremal questions play a major role in the study of communication network vulnerability. These extremal problems are special cases of questions concerning the realizability of graph invariants. We define a CS(\(p, q, \lambda, \delta\)) graph as a connected, separable graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). In this notation, if the “CS” is omitted the graph is not necessarily connected and separable. An arbitrary quadruple of integers \((a, b, c, d)\) is called CS(\(p, q, A, 5\)) realizable if there is a CS(\(p, q, \lambda, \delta\)) graph with \(p = a, q = b, \lambda = c\) and \(\delta= d\). Necessary and sufficient conditions for a quadruple to be CS(\(p, q,\lambda, \delta\)) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta), (p, q, \lambda, \Delta), (p, q, \delta, \Delta), (p, q, \lambda, \delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(A\) denotes the maximum degree for all points in a graph and \(\lambda\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa), (p, q, \lambda), (p, q, \delta), (p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.

We use \(k\)-trees to generalize the sequence of Motzkin numbers and show that Baxter’s generalization of Temperley-Lieb operators is a special case of our generalization of Motzkin numbers. We also obtain a recursive summation formula for the terms of \(3\)-Motzkin numbers and investigate some asymptotic properties of the terms of \(k\)-Motzkin numbers.

In this article, defining the matrix extensions of the Fibonacci and Lucas numbers, we start a new approach to derive formulas for some integer numbers which have appeared, often surprisingly, as answers to intricate problems, in conventional and in recreational Mathematics. Our approach provides a new way of looking at integer sequences from the perspective of matrix algebra, showing how several of these integer sequences relate to each other.

For a finite group \(G\) the commutativity degree,

\[d(G)=\frac{|\{(x,y)|x,y \in G, xy=yx\}|}{|G|^2}\]

is defined and studied by several authors and when \(d(G) \geq \frac{1}{2}\) it is proved by P. Lescot in 1995 that \(G\) is abelian , or \(\frac{G}{Z(G)}\) is elementary abelian with \(|G’| = 2\), or \(G\) is isoclinic with \(S_3\) and \(d(G) = 1\). The case when \(d(G) < \frac{1}{2}\) is of interest to study. In this paper we study certain infinite classes of finite groups and give explicit formulas for \(d(G)\). In some cases the groups satisfy \(\frac{1}{4} < d(G) < \frac{1}{2}\). Some of the groups under study are nilpotent of high nilpotency classes.