We study the spectral radius of graphs with \(n\) vertices and a \(k\)-vertex cut and describe the graph which has the maximal spectral radius in this class. We also discuss the limit point of the maximal spectral radius.

Consider lattice paths in \(\mathbb{Z}^2\) taking unit steps north (N) and east (E). Fix positive integers \(r,s\) and put an equivalence relation on points of \(\mathbb{Z}^2\) by letting \(v,w\) be equivalent if \(v-w = \ell(r,s)\) for some \(k \in \mathbb{Z}\). Call a lattice path \({valid}\) if whenever it enters a point \(v\) with an E-step, then any further points of the path in the equivalence class of \(v\) are also entered with an E-step. Loehr and Warrington conjectured that the number of valid paths from \((0,0)\) to \((nr,ns)\) is \({\binom{r+s}{nr}}^n\). We prove this conjecture when \(s=2\).

Given integers \(m \geq 2, r \geq 2\), let \(q_m(n), q_0^{(m)}(n), b_r^{(m)}(n)\) denote respectively the number of \(m\)-colored partitions of \(n\) into: distinct parts, distinct odd parts, and parts not divisible by \(r\).We obtain recurrences for each of the above-mentioned types of partition functions.

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.