A digraph \(D\) is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, \(n\). Corresponding results for biclique covers and partitions of the digraph are provided.

The current paper deals with two special matrices \(T_n\) and \(W_n\) related to the Pascal, Vandermonde, and Stirling matrices. As a result, various properties of the entries of \(T_n\) and \(W_n\) are obtained, including the generating functions, recurrence relations, and explicit expressions. Some additional results are also presented.

There are some results and many conjectures with the conclusion that a graph \(G\) contains all trees of given size \(k\). We prove some new results of this type.

In \([3]\), we gave a factorization of the generalized Lah matrix.In this short note, we show its another factorization. From this factorization, several interesting combinatorial identities involving the Fibonacci numbers are obtained.

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-decomposition of \(K_v\), denoted by \(G-GD_\lambda(v)\), 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 exactly one block of \(\mathcal{B}\). In this paper, nine graphs \(G_i\) with six vertices and nine edges are discussed, and the existence of \(G_i-GD_\lambda(v)\) is given, \(1 \leq i \leq 9\).

Let \(G = (V, E)\) be a graph. A set \(S \subseteq V\) is a restrained dominating set 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 smallest cardinality of a restrained dominating set of \(G\). It is known that if \(T\) is a tree of order \(n\), then \(\gamma_r(T) \geq \left\lceil \frac{n+2}{3} \right\rceil\). In this note, we provide a simple constructive characterization of the extremal trees \(T\) of order \(n\) achieving this lower bound.

Given non-negative integers \(r, s\), and \(t\), an \([r, s, t]\)-coloring of a graph \(G = (V(G), E(G))\) is a mapping \(c\) from \(V(G) \cup E(G)\) to the color set \(\{0, 1, \ldots, k-1\}\) such that \(|c(v_i) – c(v_j)| \geq r\) for every two adjacent vertices \(v_i, v_j\), \(|c(e_i) – c(e_j)| \geq s\) for every two adjacent edges \(e_i, e_j\), and \(|c(v_i) – c(e_i)| \geq t\) for all pairs of incident vertices and edges, respectively. The \([r, s, t]\)-chromatic number \(\chi_{r,s,t}(G)\) of \(G\) is defined to be the minimum \(k\) such that \(G\) admits an \([r, s, t]\)-coloring. We prove that \(\chi_{1,1,2}(K_5) = 7\) and \(\chi_{1,1,2}(K_6) = 8\).

We determine a recursive formula for the number of rooted complete \(N\)-ary trees with \(n\) leaves, which generalizes the formula for the sequence of Wedderburn-Etherington numbers. The diagonal sequence of our new sequences equals the sequence of numbers of rooted trees with \(N + 1\) vertices.

In this paper, we determine the conics characterizing the generalized Fibonacci and Lucas sequences with indices in arithmetic progressions, generalizing work of Melham and McDaniel.

A graph \(G = (V, E)\) is a mod sum graph if there exists a positive integer \(z\) and a labeling, \(\lambda\), of the vertices of \(G\) with distinct elements from \(\{1, 2, \ldots, z-1\}\) such that \(uv \in E\) if and only if the sum, modulo \(z\), of the labels assigned to \(u\) and \(v\) is the label of a vertex of \(G\). The mod sum number \(\rho(G)\) of a connected graph \(G\) is the smallest nonnegative integer \(m\) such that \(G \cup mK_1\), the union of \(G\) and \(m\) isolated vertices, is a mod sum graph. In Section \(2\), we prove that \(F_n\) is not a mod sum graph and give the mod sum number of \(F_n\) (\(n \geq 6\) is even). In Section \(3\), we give the mod sum number of the symmetric complete graph.