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.

Let \(\{w_n\}\) be a second-order recurrent sequence. Several identities about the sums of products of second-order recurrent sequences were obtained and the relationship between the second-order recurrent sequences and the recurrence coefficient revealed. Some identities about Lucas sequences, Lucas numbers, and Fibonacci numbers were also obtained.

In this paper, we prove that for any positive integers \(k,n\) with \(k \geq 2\) , the graph \(P_k^n\) is a divisor graph if and only if \(n \leq 2k + 2\) , where \(P^k_n\) is the \(k\) th power of the path \(P_n\). For powers of cycles we show that \(C^k_n\) is a divisor graph when \(n \leq 2k + 2\), but is not a divisor graph when \(n \geq 2k + 2\),but is not a divisor graph when \(n\geq 2k+\lfloor \frac{k}{2}\rceil,\) where \(C^k_n\) is the \(k\)th power of the cycle \(C_n\). Moreover, for odd \(n\) with \(2k+2 < n < 2k + \lfloor\frac{k}{2}\rfloor + 3\), we show that the graph \(C^k_n\) is not a divisor graph.

The Wiener index of a graph \(G\) is defined as \(W(G) = \sum_{u,v \in V(G)} d_G(u,v),\) where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\) and the sum goes over all pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the minimal and maximal Wiener index.