This paper is an extension of the work [On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math.
and Comp., \(160 (2005), 125-132.]\), in which for some norms of the circulant matrices with classical Fibonacci and Lucas numbers it is
obtained the lower and upper bounds. In this new paper, we generalize the results of that work.

Let \(a_0, a_1, \ldots, a_{r-1}\) be positive integers and define a conditional sequence \(\{q_n\}\), with initial conditions \(q_0 = 0\) and \(q_1 = 1\), and for all \(n \geq 2\), \(q_n = a_1q_{n-1} + q_{n-2}\) where \(n \equiv t \pmod{r}\). For \(r = 2\), the author studied it in \([1]\). For general \(\{q_n\}\), we found a closed form of the generating function for \(\{q_n\}\) in terms of the continuant in \([2]\). In this paper, we give the matrix representation and a Binet-like formula for the conditional sequence \(\{q_n\}\) by using the matrix methods.

A locally \(nK_2\) graph \(G\) is a graph such that the set of neighbors of any vertex of \(G\) induces a subgraph isomorphic to \(nK_2\). We show that a locally \(nK_2\) graph \(G\) must have at least \(6n – 3\) vertices, and that a locally \(nK_2\) graph with \(6n – 3\) vertices exists if and only if \(n \in \{1, 2, 3, 5\}\), and in these cases the graph is unique up to isomorphism. The case \(n = 5\) is surprisingly connected to a classic theorem of algebraic geometry: The only locally \(5K_2\) graph on \(6 \times 5 – 3 = 27\) vertices is the incidence graph of the 27 straight lines on any nonsingular complex projective cubic surface.

Every graph can be associated to a cheracteristic exponential equation involving powers of (say) \(2\), whose unknowns represent ver-
tex labels and whose general solution is equivalent to a graceful labelling of the graph. If we do not require that the solutions be
integers, we obtain a generalisation of a graceful labelling that uses real numbers as labels. Some graphs that are well known to be non-graceful become graceful in this more general context. Among other things, “real-graceful” labellings provide some information on the Tigidity to be non-graceful, also asymptotically.

Let \(G\) be a graph of order \(n\), and let \(a, b, k\) be nonnegative integers with \(1 \leq a \leq b\). A spanning subgraph \(F\) of \(G\) is called an \([a, b]\)-factor if \(a \leq d_F(x) \leq b\) for each \(x \in V(G)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if \(G – N\) has an \([a, b]\)-factor for each \(N \subseteq V(G)\) with \(|N| = k\). In this paper, it is proved that \(G\) is an \((a, b, k)\)-critical graph if \(n \geq \frac{(a+b-1)(a+b-2)}{b} +\frac{bk}{b-1}\), \(bind(G) \geq \frac{(a+b-1)(n-1)}{b(n-1-k)}\), and \(\delta(G) \neq \left\lfloor \frac{(a-1)n+a+b+bk-2}{a+b-1} \right\rfloor\).

The vulnerability shows the resistance of the network until communication breakdown after the disruption of certain stations or communication links. This study introduces a new vulnerability parameter, neighbor rupture degree. The neighbor rupture degree of a non-complete connected graph \(G\) is defined to be

\[Nr(G) = \max\{w(G/S) – |S| – c(G/S): S \subset V(G), w(G/S) \geq 1\}\]

where \(S\) is any vertex subversion strategy of \(G\), \(w(G/S)\) is the number of connected components in \(G/S\), and \(c(G/S)\) is the maximum order of the components of \(G/S\). In this paper, the neighbor rupture degree of some classes of graphs are obtained and the relations between neighbor rupture degree and other parameters are determined.

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).

In this paper, we obtain the following upper and lower bounds for \(q\)-factorial \([n]_q!\):

\[(q; q)_\infty (1 – q)^{-n} e^{f_q(n+1)} < [n]_q! < (q; q)_\infty (1 – q)^{-n} e^{g_q(n+1)},\] where \(n \geq 1\), \(0 < q < 1\), and the two sequences \(f_q(n)\) and \(g_q(n)\) tend to zero through positive values. Also, we present two examples of the two sequences \(f_q(n)\) and \(g_q(n)\).

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).