We show that every generalized quadrangle of order \((4,6)\) with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle \(\text{AS}(5)\). It then easily follows that every generalized quadrangle of order \(5\) with an axis of symmetry is isomorphic to the classical generalized quadrangle \(\text{Q}(4, 5)\).

A \(C_5C_7\) net is a trivalent decoration made by alternating pentagons \(C_5\) and heptagons \(C_7\). It can cover either a cylinder or a torus. In this paper, we compute the Szeged index of \(HC_5C_7[ r, p ]\) nanotube.

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types \(C, D\), and \(L\). Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.

The number of essentially different square polyominoes of order \(n\) and minimum perimeter \(p(n)\) is enumerated.

Let \(G = (V, E)\) be a graph. Then \(S \subseteq V\) is an excess-\(t\) global powerful alliance if \(|N[v] \cap S| \geq |N[v] \cap (V – S)| + t\) for every \(v \in V\). If \(t = 0\), this definition reduces to that of a \({global \;powerful \;alliance}\). Here we determine bounds on the cardinalities of such sets \(S\).

A total perfect code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is adjacent to exactly one vertex in the subset. We prove that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code.

We calculate the norm of weighted composition operators \(uC_\psi\) from the Bloch space to the weighted space \(H^\infty_\mu({B})\) on the unit ball \({B}\).

Let \(P\) be a polygon whose vertices have been colored (labeled) cyclically with the numbers \(1, 2, \ldots, c\). Motivated by conjectures of Propp, we are led to consider partitions of \(P\) into \(k\)-gons which are proper in the sense that each \(k\)-gon contains all \(c\) colors on its vertices. Counting the number of proper partitions involves a generalization of the \(k\)-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.

Let \(n, k\) be integers and \(k < n\). Denote by \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) the set of graphs of order \(n\) with \(k\) independent vertices and the set of graphs of order \(n\) with \(k\) independent edges, respectively. The bounds of the spectral radius of graphs in \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) are obtained.

Let \(n \in \mathbb{N}\) and let \(A \subseteq \mathbb{Z}_n\) be such that \(A\) does not contain \(0\) and is non-empty. We define \({E}_A(n)\) to be the least \(t \in \mathbb{N}\) such that for all sequences \((x_1, \ldots, x_t) \in \mathbb{Z}^t\), there exist indices \(j_1, \ldots, j_n \in \mathbb{N}\), \(1 \leq j_1 < \cdots < j_n \leq t\), and \((\theta_1, \ldots, \theta_n) \in A^n\) with \(\sum_{i=1}^n \theta_i x_{j_i} \equiv 0 \pmod{n}\). Similarly, for any such set \(A\), we define the \({Davenport Constant}\) of \(\mathbb{Z}_n\) with weight \(A\) denoted by \(D_A(n)\) to be the least natural number \(k\) such that for any sequence \((x_1, \ldots, x_k) \in \mathbb{Z}^k\), there exist a non-empty subsequence \((x_{j}, \ldots, x_{j_i})\) and \((a_1, \ldots, a_l) \in A^t\) such that \(\sum_{i=1}^n a_i x_{j_i} \equiv 0 \pmod{n}\). Das Adhikari and Rath conjectured that for any set \(A \subseteq \mathbb{Z}_n \setminus \{0\}\), the equality \({E}_A(n) = D_A(n) + n – 1\) holds. In this note, we determine some Davenport constants with weights and also prove that the conjecture holds in some special cases.