Let \(C_n^{(t)}\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 7, 9, 4k\). In this paper, the conjecture is shown to be true for \(n = 11\).

Let \(P(G; \lambda)\) denote the chromatic polynomial of a graph \(G\), expressed in the variable \(\lambda\). Then \(G\) is said to be chromatically unique if \(G\) is isomorphic with \(H\) for any graph \(H\) such that \(P(H; \lambda) = P(G; \lambda)\). The graph consisting of \(s\) edge-disjoint paths joining two vertices is called an \(s\)-bridge graph. In this paper, we provide a new family of chromatically unique \(5\)-bridge graphs.

Recently in \([5]\), the author considered certain reciprocal sums of general second order recurrence \(\{W_n\}\). In this paper, we generalize the results of Xi and we give some new results for the reciprocal sums of \(k\)-th power of general second order recurrence \(\{W_{kn}\}\) for arbitrary positive integer \(k\).

In this article, we study the generalized Bernoulli and Euler polynomials, and obtain relationships between them, based upon the technique of matrix representation.

Let \(K_v\) be the complete graph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-decomposition of \(K_v\), denoted by \(G\)-GD\((v)\), 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 \(G_i\)-decompositions are completely solved, \(1 \leq i \leq 9\).

A graph \(G\) is called \({uniquely\; k-list \;colorable}\), or \(UkLC\) for short, if it admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring. A graph \(G\) is said to have the property \(M(k)\) (M for Marshal Hall) if and only if it is not \(UkLC\). In \(1999\), M. Ghebleh and E.S. Mahmoodian characterized the \(U3LC\) graphs for complete multipartite graphs except for nine graphs. At the same time, for the nine exempted graphs, they give an open problem: verify the property \(M(3)\) for the graphs \(K_{2,2,\ldots,2}\) for \(r = 4,5,\ldots,8\), \(K_{2,3,4}\), \(K_{1*4,4}\), \(K_{1*4,4}\), and \(K_{1*5,4}\). Until now, except for \(K_{1*5,4}\), the other eight graphs have been showed to have the property \(M(3)\) by W. He et al. In this paper, we show that graph \(K_{1*5,4}\) has the property \(M(3)\), and as consequences, \(K_{1*4,4}\), \(K_{2,2,4}\) have the property \(M(3)\). Therefore the \(U3LC\) complete multipartite graphs are completely characterized.

Given a graph \(G\), we say \(S \subseteq V(G)\) is \({resolving}\) if for each pair of distinct \(u, v \in V(G)\) there is a vertex \(x \in S\) where \(d(u, x) \neq d(v, x)\). The metric dimension of \(G\) is the minimum cardinality of all resolving sets. For \(w \in V(G)\), the distance from \(w\) to \(S\), denoted \(d(w, S)\), is the minimum distance between \(w\) and the vertices of \(S\). Given \(\mathcal{P} = \{P_1, P_2, \ldots, P_k\}\) an ordered partition of \(V(G)\), we say \(P\) is resolving if for each pair of distinct \(u, v \in V(G)\) there is a part \(P_i\) where \(d(u, P_i) \neq d(v, P_i)\). The partition dimension is the minimum order of all resolving partitions. In this paper, we consider relationships between metric dimension, partition dimension, diameter, and other graph parameters. We construct “universal examples” of graphs with given partition dimension, and we use these to provide bounds on various graph parameters based on metric and partition dimensions. We form a construction showing that for all integers \(a\) and \(b\) with \(3 \leq a \leq \beta + 1\), there exists a graph \(G\) with partition dimension \(\alpha\) and metric dimension \(\beta\), answering a question of Chartrand, Salehi, and Zhang \([3]\).

The total chromatic number \(\chi_\tau(G)\) is the least number of colours needed to colour the vertices and edges of a graph \(G\) such that no incident or adjacent elements (vertices or edges) receive the same colour. This work determines the total chromatic number of grids, particular cases of partial grids, near-ladders, and of \(k\)-dimensional cubes.

The \({star arboricity}\) \(sa(G)\) of a graph \(G\) is the minimum number of star forests which are needed to decompose all edges of \(G\). For integers \(k\) and \(n\), \(1 \leq k \leq n\), the \({crown}\) \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : i = 0, 1, \ldots, n-1, j \equiv i+1, i+2, \ldots, i+k \pmod{n}\}\). In \([2]\), Lin et al. conjectured that for every \(k\) and \(n\), \(3 \leq k \leq n-1\), the star arboricity of the crown \(C_{n,k}\) is \(\lceil k/2 \rceil + 1\) if \(k\) is odd and \(\lceil k/2 \rceil + 2\) otherwise. In this note, we show that the above conjecture is not true for the case \(n = 9t\) (\(t\) is a positive integer) and \(k = 4\) by showing that \(sa(C_{9t,4}) = 3\).

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 \(\text{Q}(4, 5)\).

A \(C_5C_7\) net is a trivalent decoration made by alternating pentagons \(C_5\) and heptagons \(C_7\). It can cover either a cylinder or a torus. In this paper, we compute the Szeged index of \(HC_5C_7[ r, p ]\) nanotube.

We present algebraic constructions yielding incidence matrices for all finite Desarguesian elliptic semiplanes of types \(C, D\), and \(L\). Both basic ingredients and suitable notations are derived from addition and multiplication tables of finite fields. This approach applies also to the only elliptic semiplane of type B known so far. In particular, the constructions provide intrinsic tactical decompositions and partitions for these elliptic semiplanes into elliptic semiplanes of smaller order.

The number of essentially different square polyominoes of order \(n\) and minimum perimeter \(p(n)\) is enumerated.

Let \(G = (V, E)\) be a graph. Then \(S \subseteq V\) is an excess-\(t\) global powerful alliance if \(|N[v] \cap S| \geq |N[v] \cap (V – S)| + t\) for every \(v \in V\). If \(t = 0\), this definition reduces to that of a \({global \;powerful \;alliance}\). Here we determine bounds on the cardinalities of such sets \(S\).

A total perfect code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is adjacent to exactly one vertex in the subset. We prove that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code.

We calculate the norm of weighted composition operators \(uC_\psi\) from the Bloch space to the weighted space \(H^\infty_\mu({B})\) on the unit ball \({B}\).

Let \(P\) be a polygon whose vertices have been colored (labeled) cyclically with the numbers \(1, 2, \ldots, c\). Motivated by conjectures of Propp, we are led to consider partitions of \(P\) into \(k\)-gons which are proper in the sense that each \(k\)-gon contains all \(c\) colors on its vertices. Counting the number of proper partitions involves a generalization of the \(k\)-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.

Let \(n, k\) be integers and \(k < n\). Denote by \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) the set of graphs of order \(n\) with \(k\) independent vertices and the set of graphs of order \(n\) with \(k\) independent edges, respectively. The bounds of the spectral radius of graphs in \(\mathcal{G}_{n,k}\) and \(\mathcal{G}'_{n,k}\) are obtained.

Let \(n \in \mathbb{N}\) and let \(A \subseteq \mathbb{Z}_n\) be such that \(A\) does not contain \(0\) and is non-empty. We define \({E}_A(n)\) to be the least \(t \in \mathbb{N}\) such that for all sequences \((x_1, \ldots, x_t) \in \mathbb{Z}^t\), there exist indices \(j_1, \ldots, j_n \in \mathbb{N}\), \(1 \leq j_1 < \cdots < j_n \leq t\), and \((\theta_1, \ldots, \theta_n) \in A^n\) with \(\sum_{i=1}^n \theta_i x_{j_i} \equiv 0 \pmod{n}\). Similarly, for any such set \(A\), we define the \({Davenport Constant}\) of \(\mathbb{Z}_n\) with weight \(A\) denoted by \(D_A(n)\) to be the least natural number \(k\) such that for any sequence \((x_1, \ldots, x_k) \in \mathbb{Z}^k\), there exist a non-empty subsequence \((x_{j}, \ldots, x_{j_i})\) and \((a_1, \ldots, a_l) \in A^t\) such that \(\sum_{i=1}^n a_i x_{j_i} \equiv 0 \pmod{n}\). Das Adhikari and Rath conjectured that for any set \(A \subseteq \mathbb{Z}_n \setminus \{0\}\), the equality \({E}_A(n) = D_A(n) + n – 1\) holds. In this note, we determine some Davenport constants with weights and also prove that the conjecture holds in some special cases.

In this paper, we introduce an extension of the hyperbolic Fibonacci and Lucas functions which were studied by Stakhov and Rozin. Namely, we define hyperbolic functions by second-order recurrence sequences and study their hyperbolic and recurrence properties. We give the corollaries for Fibonacci, Lucas, Pell, and Pell-Lucas numbers. We finalize with the introduction of some surfaces (the Metallic Shofars) that relate to the hyperbolic functions with the second-order recurrence sequences.

The graph’s irregularity is the sum of the absolute values of the differences of degrees of pairs of adjacent vertices in the graph. We provide various upper bounds for the irregularity of a graph, especially for \(K_{r+1}\)-free graphs, where \(K_{r+1}\) is a complete graph on \(r+1\) vertices, and trees and unicyclic graphs of given number of pendant vertices.

Let \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) be the \(n\)-dimensional vector (resp. affine) space over the finite field \(\mathbb{F}_q\). For \(1 \leq i \leq i+s \leq n-1\) (resp. \(0 \leq i \leq i+s \leq n-1\)), let \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) denote the set of all subspaces (resp. flats) in \(\mathbb{F}_q^(n)\) (resp. \({AG}(n,\mathbb{F}_q)\)) with dimensions between \(i\) and \(i+s\) including \(\mathbb{F}_q^(n)\) and \(\{0\}\) (resp. \(\emptyset\)). By ordering \(\mathcal{L}(i,i+s;n)\) (resp. \(\mathcal{L}'(i,i+s;n)\)) by ordinary inclusion or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity.

In this paper, we give some relations involving the usual Fibonacci and generalized order-\(k\) Pell numbers. These relations show that the generalized order-\(k\) Pell numbers can be expressed as the summation of the usual Fibonacci numbers. We find families of Hessenberg matrices such that the permanents of these matrices are the usual Fibonacci numbers, \(F_{2i-1}\), and their sums. Also, extending these matrix representations, we find families of super-diagonal matrices such that the permanents of these matrices are the generalized order-\(k\) Pell numbers and their sums.

Let \(G\) be a finite group and \(S\) be a subset (possibly containing the identity element) of \(G\). We define the Bi-Cayley graph \(X = BC(G, S)\) to be the bipartite graph with vertices \(G \times \{0, 1\}\) and edges \(\{(g, 0), (sg, 1) : g \in G, s \in S\}\). In this paper, we show that if \(X = BC(G, S)\) is connected, then \(\kappa(X) = \delta(X)\).

Some new characterizations for harmonic Bergman space on the unit ball \({B}\) in \(\mathbb{R}^n\) are given in this paper. They can be described as derivative-free characterizations.

The planar Ramsey number \(PR(H_1, H_2)\) is the smallest integer \(n\) such that any planar graph on \(n\) vertices contains a copy of \(H_1\) or its complement contains a copy of \(H_2\). It is known that the Ramsey number \(R(K_4 – e, K_k – e)\) for \(k \leq 6\). In this paper, we prove that \(PR(K_4 – e, K_6 – e) = 16\) and show the lower bounds on \(PR(K_4 – e, K_k – e)\).