We estimate the essential norm of the weighted composition operator \(uC_{\varphi}\) from the weighted Bergman space \(A^{p}_{\alpha}(\mathbb{B})\) to the weighted space \(H^{\infty}_{\mu}(\mathbb{B})\) on the unit ball \(\mathbb{B}\), when \(p > 1\) and \(\alpha \geq -1\) (for \(\alpha = -1\), \(A^{p}_{\alpha}\) is the Hardy space \(H^{p}(\mathbb{B})\)). We also give a necessary and sufficient condition for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu}(B)\) to be compact, and for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu,0}(\mathbb{B})\) to be bounded or compact, when \(p > 0\), \(\alpha \geq -1\).

Let \(G = (V,E)\) be a graph. A set \(S \subseteq V\) is called a restrained dominating set of \(G\) if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V – S\). The restrained domination number of \(G\), denoted by \(\gamma_r(G)\), is the minimum cardinality of a restrained dominating set of \(G\). In this paper, we establish an upper bound on \(\gamma_r(G)\) for a connected graph \(G\) by the probabilistic method.

Any vertex labeling \(f: V \to \{0,1\}\) of the graph \(G = (V,E)\) induces a partial edge labeling \(f^*: E \to \{0,1\}\) defined by \(f^*(uv) = f(u)\) if and only if \(f(u) = f(v)\). The balance index set of \(G\) is defined as \(\{|f^{*{-1}}(0) – f^{*{-1}}(1)|: |f^{-1}(0) – f^{-1}(1)| \leq 1\}\). In this paper, we first determine the balance index sets of rooted trees of height not exceeding two, thereby completely settling the problem for trees with diameter at most four. Next we show how to extend the technique to rooted trees of any height, which allows us to derive a method for determining the balance index set of any tree.

We show that partial permutation decoding can be used, and give explicit \(s\)-PD-sets in the symmetric group, where \(s\) is less than the full error-correction capability of the code, for some classes of binary codes obtained from the adjacency matrices of the graphs with vertices the \(\binom{n}{3}\) \(3\)-subsets of a set of size \(n\) with adjacency defined by the vertices as \(3\)-sets being adjacent if they have a fixed number of elements in common.

Let \(G\) be a simple connected graph. For a subset \(S\) of \(V(G)\) with \(|S| = 2n + 1\), let \(\alpha_{(2n+1)}(G,S)\) denote the graph obtained from \(G\) by contracting \(S\) to a single vertex. The graph \(\alpha_{(2n+1)}(G, S)\) is also said to be obtained from \(G\) by an \(\alpha_{(2n+1)}\)-contraction. For pairwise disjoint subsets \(S_1, S_2, \ldots, S_{2n}\) of \(V(G)\), let \(\beta_n(G, S_1, S_2, \ldots, S_{2n})\) denote the graph obtained from \(G\) by contracting each \(S_i\) (\(i = 1, 2, \ldots, 2n\)) to a single vertex respectively. The graph \(\beta_{2n}(G, S_1, S_2, \ldots, S_{2n})\) is also said to be obtained from \(G\) by a \(\beta_{2n}\)-contraction. In the present paper, based on \(\alpha_{(2n+1)}\)-contraction and \(\beta_{2}\)-contraction, some new characterizations for \(n\)-extendable bipartite graphs are given.

A graph \(G\) is quasi-claw-free if it satisfies the property: \(d(x, y) = 2 \Rightarrow\) there exists \(u \in N(x) \cap N(y)\) such that \(N[u] \subseteq N[x] \cup N[y]\). In this paper, we prove that the circumference of a \(2\)-connected quasi-claw-free graph \(G\) on \(n\) vertices is at least \(\min\{3\delta + 2, n\}\) or \(G \in \mathcal{F}\), where \(\mathcal{F}\) is a class of nonhamiltonian graphs of connectivity \(2\). Moreover, we prove that if \(n \leq 40\), then \(G\) is hamiltonian or \(G \in \mathcal{F}\).

Let \(K_{n,n}\) denote the complete bipartite graph with \(n\) vertices in each part. In this paper, it is proved that there is no cyclic \(m\)-cycle system of \(K_{n,n}\) for \(m \equiv 2 \pmod{4}\) and \(n \equiv 2 \pmod{4}\). As a consequence, necessary and sufficient conditions are determined for the existence of cyclic \(m\)-cycle systems of \(K_{n,n}\) for all integers \(m \leq 30\).

We examine a design \(\mathcal{D}\) and a binary code \(C\) constructed from a primitive permutation representation of degree \(2025\) of the sporadic simple group \(M^c L\). We prove that \(\text{Aut}(C) = \text{Aut}(\mathcal{D}) = M^c L\) and determine the weight distribution of the code and that of its dual. In Section \(6\) we show that for a word \(w_i\) of weight \(7\), where \(i \in \{848, 896, 912, 972, 1068, 1100, 1232, 1296\}\) the stabilizer \((M^\circ L)_{w_i}\) is a maximal subgroup of \(M^\circ L\). The words of weight \(1024\) split into two orbits \(C_{(1024)_1}\) and \(C_{(1024)_2}\), respectively. For \(w_i \in C_{(1024)_1}\), we prove that \((M^c L)_{w_i}\) is a maximal subgroup of \(M^c L\).

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v, G, \lambda)\)-PD \(((v, G,\lambda)\)-CD), is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(\mathcal{B}\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we have completely determined the packing number and covering number for the graphs with seven points, seven edges and an even cycle.

In this paper, it is shown that there are exactly \(5\) non-isomorphic abstract ovals of order \(9\), all of them projective. The result has been obtained via an exhaustive search, based on the classification of the \(1\)-factorizations of the complete graph with \(10\) vertices.