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.

In this paper, we consider the effect of edge contraction on the domination number and total domination number of a graph. We define the (total) domination contraction number of a graph as the minimum number of edges that must be contracted in order to decrease the (total) domination number. We show that both of these two numbers are at most three for any graph. In view of this result, we classify graphs by their (total) domination contraction numbers and characterize these classes of graphs.

In this paper, connected graphs with the largest Laplacian eigenvalue at most \(\frac{5+\sqrt{13}}{2}\) are characterized. Moreover, we prove that these graphs are determined by their Laplacian spectrum.

An extended directed triple system of order \(v\) with an idempotent element (EDTS(\(v, a\))) is a collection of triples of the type \([x, y, z]\), \([x, y, x]\) or \((x, x, x)\) chosen from a \(v\)-set, such that every ordered pair (not necessarily distinct) belongs to only one triple and there are \(a\) triples of the type \((x, x, x)\). If such a design with parameters \(v\) and \(a\) exists, then it will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). A necessary and sufficient condition for the existence of EDTS(\(v, 0\)) and EDTS(\(v, 1\)) are \(v \equiv 0 \pmod{3}\) and \(v \not\equiv 0 \pmod{3}\), respectively. In this paper, we have constructed two EDTS(\(v, a\))’s such that the number of common triples is in the set \(\{0, 1, 2, \ldots, b_{v,a} – 2, b_{v,a}\}\), for \(a = 0, 1\).

As applications of the Anzahl theorems in finite orthogonal spaces, we study the critical problem of totally isotropic subspaces, and obtain the critical exponent.

Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H,\lambda) = P(G, \lambda)\) implies H is isomorphic to \(G\). In this paper, we study the chromaticity of Turén graphs with deleted edges that induce a matching or a star. As a by-product, we obtain new families of chromatically unique graphs.