Let \(T_n\) denote a complete binary tree of depth \(n\). Each internal node \(v\) of \(T_n\) has two children denoted by \(\text{left}(v)\) and \(\text{right}(v)\). Let \(f\) be a function mapping each internal node \(v\) to \(\{\text{left}(v), \text{right}(v)\}\). This naturally defines a path from the root, \(\lambda\), of \(T_n\) to one of its leaves given by

\[\lambda, f(\lambda), f^2(\lambda), \ldots f^n(\lambda).\]

We consider the problem of finding this path via a deterministic algorithm that probes the values of \(f\) in parallel. We show that any algorithm that probes \(k\) values of \(f\) in one round requires \(\frac{n}{\lfloor \log(k+1) \rfloor}\) rounds in the worst case. This indicates that the amount of information that can be extracted in parallel is, at times, strictly less than the amount of information that can be extracted sequentially.

A graph \(G\) is edge-\(L\)-colorable, if for a given edge assignment \(L = \{L(e) : e \in E(G)\}\), there exists a proper edge-coloring \(\phi\) of \(G\) such that \(\phi(e) \in L(e)\) for all \(e \in E(G)\). If \(G\) is edge-\(L\)-colorable for every edge assignment \(L\) with \(|L(e)| \geq k\) for \(e \in E(G)\), then \(G\) is said to be edge-\(k\)-choosable. In this paper, we prove that if \(G\) is a planar graph without chordal \(7\)-cycles, then \(G\) is edge-\(k\)-choosable, where \(k = \max\{8, \Delta(G) + 1\}\).

In this note, we study some properties of the composition operator \(C_\varphi\) on the Fock space \(\mathcal{F}_X^2\) of \(X\)-valued analytic functions in \(\mathbb{C}\). We give a necessary and sufficient condition for a bounded operator on \(\mathcal{F}_X^2\) to be a composition operator and for the adjoint operator of a composition operator to be also a composition operator on \(\mathcal{F}_X^2\). We also give characterizations of normal, unitary, and co-isometric composition operators on \(\mathcal{F}_X^2\).

The competition hypergraph \(C\mathcal{H}(D)\) of a digraph \(D\) is the hypergraph such that the vertex set is the same as \(D\) and \(e \subseteq V(D)\) is a hyperedge if and only if \(e\) contains at least \(2\) vertices and \(e\) coincides with the in-neighborhood of some vertex \(v\) in the digraph \(D\). Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number \(hk(\mathcal{H})\) of a hypergraph \(\mathcal{H}\) is defined to be the smallest number of such isolated vertices.

In this paper, we study the hypercompetition numbers of hypergraphs. First, we give two lower bounds for the hypercompetition numbers which hold for any hypergraphs. And then, by using these results, we give the exact hypercompetition numbers for some family of uniform hypergraphs. In particular, we give the exact value of the hypercompetition number of a connected graph.

In this paper, we study the signed and minus total domination problems for two subclasses of bipartite graphs: biconvex bipartite graphs and planar bipartite graphs. We present a unified method to solve the signed and minus total domination problems for biconvex bipartite graphs in \(O(n + m)\) time. We also prove that the decision problem corresponding to the signed (respectively, minus) total domination problem is NP-complete for planar bipartite graphs of maximum degree \(3\) (respectively, maximum degree \(4\)).

The edge versions of Wiener index, which were based on distance between two edges in a connected graph \(G\), were introduced by Iranmanesh et al. in \(2008\). In this paper, we find the edge Wiener indices of the sum of graphs. Then as an application of our results, we find the edge Wiener indices of graphene, \(C_4\)-nanotubes and \(C_4\)-nanotori.

Let \(\kappa(G)\) be the connectivity of \(G\) and \(G \times H\) the direct product of \(G\) and \(H\). We prove that for any graphs \(G\) and \(K\), with \(n \geq 3\),\(\kappa(G \times K_n) = \min\{n\kappa(G), (n-1)\delta(G)\},\) which was conjectured by Guji and Vumar.