Let \(K_{m} – H\) be the graph obtained from \(K_{m}\) by removing the edges set \(E(H)\) of the graph \(H\) (\(H\) is a subgraph of \(K_{m}\)). We use the symbol \(Z_4\) to denote \(K_4 – P_2\). A sequence \(S\) is potentially \(K_{m} – H\)-graphical if it has a realization containing a \(K_{m} – H\) as a subgraph. Let \(\sigma(K_{m} – H, n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma(S) \geq \sigma(K_{m} – H, n)\) is potentially \(K_{m} – H\)-graphical. In this paper, we determine the values of \(\sigma(K_{r+1} – Z, n)\) for \(n \geq 5r+19, r+1 \geq k \geq 5, j \geq 5\) where \(Z\) is a graph on \(k\) vertices and \(j\) edges which contains a graph \(Z_4\), but not contains a cycle on \(4\) vertices. We also determine the values of \(\sigma(K_{r+1} – Z_4, n)\), \(\sigma(K_{r+1} – (K_4 – e), n)\), \(\sigma(K_{r+1} – K_4, n)\) for \(n \geq 5r+16, r \geq 4\).

A nowhere-zero \(k\)-tension on a graph \(G\) is a mapping from the edges of \(G\) to the set \(\{\pm 1,\pm 2,\ldots,\pm (k-1)\} \subset \mathbb{Z}\) such that, in any fixed orientation of \(G\), for each circuit \(C\) the sum of the labels over the edges of \(C\) oriented in one direction equals the sum of values of the edges of \(C\) oriented oppositely. We show that the existence of an integral tension polynomial that counts nowhere-zero \(k\)-tension on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for tensions on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero tensions on signed graphs with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the tension counting functions, and reciprocity laws that interpret the evaluations of the tension polynomials at negative integers in terms of the combinatorics of the graph.

This paper considered the concepts of monophonic, closed monophonic, and minimal closed monophonic numbers of a connected graph \(G\). It was shown that any positive integers \(m, n, d\), and \(k\) satisfying the conditions that \(4 \leq n \leq m, 3 \leq d \leq k\), and \(k \geq 2m – n + d + 1\) are realizable as the monophonic number, closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Also, any positive integers \(n, m, d\), and \(k\) with \(2 \leq n \leq m, d \geq 3\), and \(k \geq m + d – 1\) are realizable as the closed monophonic number, minimal closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Further, the closed monophonic number of the composition of connected graphs was also determined.

Let \(\Delta(G)\) be the maximum degree of a graph \(G\), and let \(\mathcal{U}(n, \Delta)\) be the set of all unicyclic graphs on \(n\) vertices with fixed maximum degree \(\Delta\). Among all the graphs in \(\mathcal{U}(n, \Delta)\) (\(\Delta \geq \frac{n+3}{2}\)), we characterize the graph with the maximal spectral radius. We also prove that the spectral radius of a unicyclic graph \(G\) on \(n\) (\(n \geq 30\)) vertices strictly increases with its maximum degree when \(\Delta(G) \geq \lceil\frac{7n}{9}\rceil + 1\).

Let \(G\) be a graph, and let \(a\), \(b\) and \(k\) be nonnegative integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of graph \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(x \in V(G)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if after any \(k\) vertices of \(G\) are deleted the remaining subgraph has an \([a, b]\)-factor. In this paper, three sufficient conditions for graphs to be \((a, b, k)\)-critical graphs are given. Furthermore, it is shown that the results in this paper are best possible in some sense.

A total dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\). A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) is dominated by at least two vertices of \(D\). The total (double, respectively) domination number of a graph \(G\) is the minimum cardinality of a total (double, respectively) dominating set of \(G\). We characterize all trees with double domination number equal to total domination number plus one.

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.