Let \(\mathcal{P}(n,k)\) denote the number of graphs on \(n+k\) vertices that contain \(P_n\), a path on \(n\) vertices, as an induced subgraph. In this note, we will find upper and lower bounds for \(\mathcal{P}(n,k)\). Using these bounds, we show that for \(k\) fixed, \(\mathcal{P}(n,k)\) behaves roughly like an exponential function of \(n\) as \(n\) gets large.

A \({dominating \;broadcast}\) of a graph \(G\) of diameter \(d\) is a function \(f: V(G) \to \{0, 1, 2, \ldots, d\}\) such that for all \(v \in V(G)\) there exists \(u \in V(G)\) with \(d(u, v) \leq f(u)\). We investigate dominating broadcasts for caterpillars.

In this paper, we obtain a fundamental result on the dynamical behavior of symmetric weighted mappings for two-dimensional real sequence spaces \({R}_s\).

In \(2006\), Mojdeh and Jafari Rad [On the total domination critical graphs, Electronic Notes in Discrete Mathematics, 24 (2006), 89-92] gave an open problem: Does there exist a \(3\)-\(\gamma_t\)-critical graph \(G\) of order \(\Delta(G) + 3\) with \(\Delta(G)\) odd and \(\delta(G) \geq 2\)? In this paper, we positively answer that for each odd integer \(n \geq 9\), there exists a \(3\)-\(\gamma_t\)-critical graph \(G_n\) of order \(n+3\) with \(\delta(G) \geq 2\). On the contrary, we also prove that for \(\Delta(G) = 3, 5, 7\), there is no \(3\)-\(\gamma_t\)-critical graph of order \(\Delta(G) + 3\) with \(\delta(G) \geq 2\).

Let \(\{w_n\}\) be a second-order recurrence sequence. According to the definition and characteristics of the recurrent sequence, we proved a recursion formula for certain reciprocal sums whose denominators are products of consecutive elements of \(\{w_n\}\).

Let \(G\) be a graph in which each vertex has been colored using one of \(k\) colors, say \(c_1, c_2, \ldots, c_k\). If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices colored \(c_i\), \(i = 1, 2, \ldots, k\), and \(|n_i – n_j| \leq 1\) for any \(i, j \in \{1, 2, \ldots, k\}\), then \(C\) is equitably \(k\)-colored. An \(m\)-cycle decomposition \(\mathcal{C}\) of a graph \(G\) is equitably \(k\)-colorable if the vertices of \(G\) can be colored so that every \(m\)-cycle in \(\mathcal{C}\) is equitably \(k\)-colored. For \(m = 4, 5\), and \(6\), we completely settle the existence problem for equitably \(2\)-colorable \(m\)-cycle decompositions of complete graphs with the edges of a \(1\)-factor added.

Suppose a network facility location problem is modelled by means of an undirected, simple graph \(G = (\mathcal{V, E})\) with \(\mathcal = \{v_1, \ldots, v_n\}\). Let \(\mathbf{r} = (r_1, \ldots, r_n)\) and \(\mathbf{s} = (s_1, \ldots, s_n)\) be vectors of nonnegative integers and consider the combinatorial optimization problem of locating the minimum number, \(\gamma(\mathbf{r}, \mathbf{s}, G)\) (say), of commodities on the vertices of \(G\) such that at least \(s_j\) commodities are located in the vicinity of (i.e. in the closed neighbourhood of) vertex \(v_j\), with no more than \(r_j\) commodities placed at vertex \(v_j\) itself, for all \(j = 1, \ldots, n\). In this paper we establish lower and upper bounds on the parameter \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a general graph \(G\). We also determine this parameter exactly for certain classes of graphs, such as paths, cycles, complete graphs, complete bipartite graphs and establish good upper bounds on \(\gamma(\mathbf{r}, \mathbf{s}, G)\) for a class of grid graphs in the special case where \(r_j = r\) and \(s_j = s\) for all \(j = 1, \ldots, n\).

Let \(A\) be an arbitrary circulant stochastic matrix, and let \(\underline{x}_0\) be a vector. An “asymptotic” canonical form is derived for \(A^k\) (as \(k \to \infty\)) as a tensor product of three simple matrices by employing a pseudo-invariant on sections of states for a Markov process with transition matrix \(A\), and by analyzing how \(A\) acts on the sections, through its auxiliary polynomial. An element-wise asymptotic characterization of \(A^k\) is also given, generalizing previous results to cover both periodic and aperiodic cases. For a particular circulant stochastic matrix, identifying the intermediate stage at which fractions first appear in the sequence \(\underline{x}_k = A^k \underline{x}_0\), is accomplished by utilizing congruential matrix identities and \((0,1)\)-matrices to determine the minimum \(2\)-adic order of the coordinates of \(\underline{x}_k\), through their binary expansions. Throughout, results are interpreted in the context of an arbitrary weighted average repeatedly applied simultaneously to each term of a finite sequence when read cyclically.

A graph \(G\) is \(s\)-Hamiltonian if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) has a Hamiltonian cycle, and \(s\)-Hamiltonian connected if for any \(S \subseteq V(G)\) of order at most \(s\), \(G-S\) is Hamiltonian-connected. Let \(k > 0, s \geq 0\) be two integers. The following are proved in this paper:(1) Let \(k \geq s+2\) and \(s \leq n-3\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s)/2\) for every independent set \(I\) of order \(k-s\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s-1\), then \(G\) is \(s\)-Hamiltonian.(2) Let \(k \geq s+3\) and \(s \leq n-2\). If \(G\) is a \(k\)-connected graph of order \(n\) and if \(\max\{d(v) : v \in I\} \geq (n+s+1)/2\) for every independent set \(I\) of order \(k-s-1\) such that \(I\) has two distinct vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha(G)+s\), then \(G\) is \(s\)-Hamiltonian connected.These results extend several former results by Dirac, Ore, Fan, and Chen.

We show that every generalized quadrangle of order \((4,6)\) with a spread of symmetry is isomorphic to the Ahrens-Szekeres generalized quadrangle \(\text{AS}(5)\). It then easily follows that every generalized quadrangle of order \(5\) with an axis of symmetry is isomorphic to the classical generalized quadrangle \(\text{Q}(4, 5)\).