An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph \(G\) with \(n\) edges that yields cyclic \(G\)-decompositions of the complete graph \(K_{2nt+1}\) was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a \(\gamma\)-labeling. Here we show that the class of almost-bipartite graphs obtained from a path with at least \(3\) edges by adding an edge joining distinct vertices of the path an even distance apart has a \(\gamma\)-labeling.

The locally twisted cube \(LTQ_n\) is an important variation of hypercube and possesses many desirable properties for interconnection networks. In this paper, we investigate the problem of embedding paths in faulty locally twisted cubes. We prove that a path of length \(l\) can be embedded between any two distinct vertices in \(LTQ_n – F\) for any faulty set \(F \subseteq V(LTQ_n) \cup E(LTQ_n)\) with \(|F| \leq n-3\) and any integer \(l\) with \(2^{n-1} \leq l \leq |V(LTQ_n – F)| – 1\) for any integer \(n > 3\). The result is tight with respect to the two bounds on path length \(l\) and faulty set size \(|F|\) for a successful embedding.

A \(G\)-design is a partition of \(E(K_v)\) in which each element induces a copy of \(G\). The existence of \(G\)-designs with the additional property that they contain no proper subsystems has been previously settled when \(G \in \{K_3, K_4 – e\}\). In this paper, the existence of \(P_m\)-designs which contain no proper subsystems is completely settled for every value of \(m\) and \(v\).

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\). In this paper, we give a sharp lower bound on the Randić index of cacti with perfect matchings.

Let \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) be the \((2v+1)\)-dimensional affine-singular symplectic space over the finite field \(\mathbb{F}_q\) and let \(\text{ASp}_{2v+1}(\mathbb{F}_q)\) be the affine-singular symplectic group of degree \(2v+1\) over \(\mathcal{F}_q\). For any orbit \(O\) of flats under \(\text{ASp}_{2v+1}(\mathbb{F}_q)\), let \(\mathcal{L}\) be the set of all flats which are intersections of flats in \(O\) such that \(O \subseteq \mathcal{L}\) and assume the intersection of the empty set of flats in \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) is \(\mathbb{F}_q^{2v+1}\). By ordering \(\mathcal{L}\) by ordinary or reverse inclusion, two lattices are obtained. This article discusses the relations between different lattices, classifies their geometricity, and computes their characteristic polynomial.

Let \(\gamma_c(G)\) be the connected domination number of \(G\).A graph is \(k\)-\(\gamma_c\)-critical if \(\gamma_c(G) = k\) and \(\gamma_c(G + uv) < \gamma_c(G)\) for any nonadjacent pair of vertices \(u\) and \(v\) in the graph \(G\). In this paper, we show that the diameter of a \(k\)-\(\gamma_c\)-critical graph is at most \(k\) and this upper bound is sharp.

A \(b\)-coloring of a graph \(G\) by \(k\) colors is a proper \(k\)-coloring of the vertices of \(G\) such that in each color class there exists a vertex having neighbors in all the other \(k-1\) color classes. The \(b\)-chromatic number \(\varphi(G)\) of a graph \(G\) is the maximum \(k\) for which \(G\) has a \(b\)-coloring by \(k\) colors. This concept was introduced by R.W. Irving and D.F. Manlove in \(1999\). In this paper, we study the \(b\)-chromatic numbers of the cartesian products of paths and cycles with complete graphs and the cartesian product of two complete graphs.

Let \(K_{d,d}\) be a complete bipartite digraph. In this paper, we determine the exact value of the domination number in iterated line digraph of \(K_{d,d}\).

A total coloring of a simple graph \(G\) is called adjacent vertex distinguishing if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) differs from the set of colors assigned to the vertices and the edges incident to \(v\). In this paper, we shall prove that the adjacent vertex distinguishing total chromatic number of an outer plane graph with \(\Delta \leq 5\) is \(\Delta+2\) if \(G\) has two adjacent maximum degree vertices, otherwise it is \(\Delta+1\).

Let \(P_j(n)\) denote the number of representations of \(n\) as a sum of \(j\) pentagonal numbers. We obtain formulas for \(P_j(n)\) when \(j = 2\) and \(j = 3\).

Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. We study some variations of this concept in which the configuration of guards induce total dominating sets. We consider two models of the problem: one in which only one guard moves at a time and one in which all guards may move simultaneously. A number of upper and lower bounds are given for the number of guards required.

Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\) then the subgraph is called a spanning subgraph of \(G\). A spanning subgraph \(H\) of \(G\) is called an \(F\)-factor if each component of \(H\) is isomorphic to \(F\). Further, if there exists a subgraph of \(G\) whose vertex set is \(V(G)\) and can be partitioned into \(F\)-factors, then it is called a \(\lambda\)-fold \(F\)-factor of \(G\), denoted by \(S_\lambda(1,F,G)\). A large set of \(\lambda\)-fold \(F\)-factors of \(G\), denoted by \(LS_\lambda(1,F,G)\), is a partition \(\{\mathcal{B}_i\}_i\) of all subgraphs of \(G\) isomorphic to \(F\), such that each \((X,\mathcal{B}_i)\) forms a \(\lambda\)-fold \(F\)-factor of \(G\). In this paper, we investigate \(LS_\lambda(1,K_{1,3},K_{v,v})\) for any index \(\lambda\) and obtain existence results for the cases \(v = 4t, 2t + 1, 12t+6\) and \(v \geq 3\).

In this paper, we give some interesting identities on the Bernoulli and the Euler numbers and polynomials by using reflection symmetric properties of Euler and Bernoulli polynomials. To derive our identities, we investigate some properties of the fermionic \(p\)-adic integrals on \(\mathbb{Z}_p\).

For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:

\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]

A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).

We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in [11] if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.

In \(1989\), Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\) and they obtained sufficient conditions for a graph to be hamiltonian with the implicit degrees. In this paper, we prove that if \(G\) is a \(2\)-connected graph of order \(n\) with \(\alpha(G) \leq n/2\) such that \(\text{id}(v) \geq (n-1)/2\) for each vertex \(v\) of \(G\), then \(G\) is hamiltonian with some exceptions.

The compact, Fredholm, and isometric weighted composition operators are characterized in this paper.

We discuss the chromaticity of one family of \(K_4\)-homeomorphs with exactly two non-adjacent paths of length two, where the other four paths are of length greater than or equal to three. We also give a sufficient and necessary condition for the graphs in the family to be chromatically unique.

In this paper, we deduced the following new Stirling series:

\[ n! \sim \sqrt{2n\pi} (\frac{n}{2})^n exp(\frac{1}{12n+1}[1 + \frac{1}{12n} (1+\frac{\frac{2}{5}}{n} + \frac{\frac{29}{150}}{n^2} – \frac{\frac{62}{2625}}{n^3} – \frac{\frac{9173}{157500}}{n^4} +\ldots )^{-1}]) ,\]

which is faster than the classical Stirling’s series.

For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:

\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]

A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).

We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in \([11]\) if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.

The general neighbor-distinguishing total chromatic number \(\chi”_{gnd}(G)\) of a graph \(G\) is the smallest integer \(k\) such that the vertices and edges of \(G\) can be colored by \(k\) colors so that no adjacent vertices have the same set of colors. It is proved in this note that \(\chi”_{gnd}(G) = \lceil \log_2 \chi(G) \rceil + 1\), where \(\chi(G)\) is the vertex chromatic number of \(G\).

A sequence \(A\) is a \(B_h^*[g]\) sequence if the coefficients of \((\sum_{a\in A}(z)^a)^h\) are bounded by \(g\). The standard Sidon sequence is a \(B[2]\) sequence. Finite Sidon sequences are called Golomb rulers, which are found to have many applications such as error correcting codes, radio frequency selection, and radio antennae placement. Let \(R_h(g,n)\) be the largest cardinality of a \(B[g]\) sequence contained in \(\{1,2,\ldots,n\}\), and \(F(h,g,k) = \min\{n : R_h(g,n) \geq k\}\). In this paper, computational techniques are applied to construct optimal generalized Sidon sequences, and \( 49\) new exact values of \(F(2,g,k)\) are found.

Recently, Chu \([5]\) derived two families of terminating \(_2F_1(2)\)-series identities. Their \(q\)-analogues will be established in this paper.

Let \(H\), \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \(G-GD_\lambda(H)\), is a partition of all the edges of \(H\) into subgraphs (called \(G\)-blocks), each of which is isomorphic to \(G\). A large set of \(G-GD_\lambda(H)\), denoted by \(G-LGD_\lambda(H)\), is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \(G-GD_\lambda(H)\)s. In this paper, we determine the existence spectrums for \(K_{2,2}-LGD_\lambda(K_{m,n})\).

A binary vertex coloring (labeling) \(f: V(G) \to \mathbb{Z}_2\) of a graph \(G\) is said to be friendly if the number of vertices labeled 0 is almost the same as the number of vertices labeled 1. This friendly labeling induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(uv) = f(u)f(v)\) for all \(uv \in E(G)\). Let \(e_f(i) = |\{uv \in E(G) : f^*(uv) = i\}|\) be the number of edges of \(G\) that are labeled \(i\). The product-cordial index of the labeling \(f\) is the number \(pc(f) = |e_f(0) – e_f(1)|\). The product-cordial set of the graph \(G\), denoted by \(PC(G)\), is defined by

\[PC(G) = \{pc(f): f \text{ is a friendly labeling of } G\}.\]

In this paper, we will determine the product-cordial sets of long grids \(P_m \times P_n\), introduce a class of fully product-cordial trees and suggest new research directions in this topic.

In this paper, we investigate some interesting identities on the Euler numbers and polynomials arising from their generating functions and difference operators. Finally, we give some properties of Bernoulli and Euler polynomials by using \(p\)-adic integral on \(\mathbb{Z}_p\).

Let \(\pi\) be a finite projective plane of order \(n\). Consider the substructure \(\pi_{n+2}\) obtained from \(\pi\) by removing \(n+2\) lines (including all points on them) no three of which are concurrent. In this paper, firstly, it is shown that \(\pi_{n+2}\) is a B-L plane and it is also homogeneous. Let \(PG(3,2)\) be a finite projective \(3\)-space of order \(n\). The substructure obtained from \(PG(3,2)\) by removing a tetrahedron that is four planes of \(PG(3,n)\) no three of which are collinear is a finite hyperbolic \(3\)-space (Olgun-Ozgir [10]). Finally, we prove that any two hyperbolic planes with the same parameters are isomorphic in this hyperbolic \(3\)-space. These results appeared in the second author’s MSc thesis.

Let \(G\) be a graph of order \(n\), and let \(a\) and \(b\) be integers such that \(1 \leq a < b\). Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) < f(x) \leq b\) for each \(x \in V(G)\). Then \(G\) has a \((g, f)\)-factor if the minimum degree \(\delta(G) \geq \frac{(b-1)^2-(a+1)(a+b-1)}{a+1}\) ,\(n>\frac{(a+b)(a+b-1)}{a+1}\) and \(\max\{d_G(x), d_G(y)\} \geq \frac{(b-1)n}{a+b}\) for any two nonadjacent vertices \(x\) and \(y\) in \(G\). Furthermore, it is shown that the result in this paper is best possible in some sense.

In this note, we consider the on-line Ramsey numbers \(\overline{R}(P_n, P_m)\) for paths. Using a high-performance computing cluster, we calculated the values for off-diagonal numbers for paths of lengths at most \(8\). Also, we were able to check that \(\overline{R}(P_9, P_9) = 17\), thus solving the problem raised in [5].

In this paper, we determine the images of hyperbolic ellipses under the Möbius and harmonic Möbius transformations.

Given a disjoint union of some complete graphs, one can define a graph by choosing one vertex from each complete graph and making all of these vertices adjacent. This observation leads us to define a new operation on certain graphs. We compute the characteristic polynomial of the resulting graphs and indicate a method for computing the determinant of this matrix for obtaining the characteristic polynomial of new graphs. We show that line graphs of trees can be obtained by performing this operation on some graphs, and, as an application, we compute the characteristic polynomials of line graphs of trees.

Consider a connected undirected graph \(G = (V, E)\) and an integer \(r \geq 1\); for any vertex \(v \in V\), let \(B_r(v)\) denote the ball of radius \(r\) centred at \(v\), i.e., the set of all vertices linked to \(v\) by a path of at most \(r\) edges. If for all vertices \(v \in V\), the sets \(B_r(v)\) are different, then we say that \(G\) is \(r\)-twin-free.

In \(r\)-twin-free graphs, we prolong the study of the extremal values that can be reached by some classical parameters in graph theory, and investigate here the maximum degree.

A new construction of authentication codes with arbitration from \((2\nu-2+2+1)\)-dimensional singular pseudo-symplectic geometry on finite fields is given. Assuming that the encoding rules are chosen according to a uniform probability distribution, the parameters and the probabilities of success for different types of deceptions are also computed.

By a defensive alliance in a graph \(G\) we mean any set \(S\) of vertices in \(G\) such that each vertex in \(S\) is adjacent to at least as many vertices inside \(S\), including the vertex itself, as outside \(S\). If, in addition, we require that every vertex outside a defensive alliance \(S\) is adjacent to at least one vertex in \(S\), then \(S\) becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of \(G\). In this paper, we determine bounds for the global defensive alliance numbers in the join, corona, and composition of graphs.

Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). A triangle is a cycle of length three. As usual, \(K_n\) denotes the complete graph on \(n\) vertices. It is shown that for all nonnegative integers \(p\) and \(q\) and for all positive integers \(n\), \(K_n\) can be decomposed into \(p\) copies of \(P_4\) and \(q\) copies of \(C_3\) if and only if \(3(p+q) = e(K_n)\), \(p \neq 1\) if \(n\) is odd, and \(p \geq \frac{n}{2}\) if \(n\) is even.

Motivated by Kotzig and Rosa’s concept of edge magic deficiency, Figueroa-Centeno, Ichishima, and Muntaner-Batle defined a similar concept for super edge magic total labelings. The super edge magic deficiency of a graph \(G\), which is denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) \(+\infty\) if there exists no such \(n\). In this paper, we study the super edge magic deficiency of kite graphs.

The corona of two graphs \(G\) and \(H\), written as \(G \odot H\), is the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and then joining the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae for the Wiener, hyper-Wiener and reverse-Wiener indices of the corona of two graphs.

The energy of a graph \(G\), denoted by \(E(G)\), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of \(G\). Let \(\mathcal{B}(p, q)\) denote the set of bipartite unicyclic graphs with a \((p, q)\)-bipartition, where \(q \geq p \geq 2\). Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. \(54 (2005) 379-388]\) conjectured that for \(q \geq 3\), \(E(B(3, q)) > E(H(3, q))\), where \(B(3, q)\) and \(H(3, q)\) are respectively graphs as shown in Fig. 1. In this note, we show that this conjecture is true for \(3 \leq q \leq 217\). As a byproduct, we determined the graph with minimal energy among all graphs in \(\mathcal{B}(3, q)\).

In this work, infinite similarities of permutation groups are investigated by means of new methods. For this purpose, we handle distinct groups on the set of natural numbers and we give the separation of the subgroups of them. Afterwards, we give the matrix representation of this groups.

This paper studies edge- and total-colorings of graphs in which (all or only adjacent) vertices are distinguished by their sets of colors. We provide bounds for the minimum number of colors needed for such colorings for the Cartesian product of graphs along with exact results for generalized hypercubes. We also present general bounds for the direct, strong and lexicographic products.

The pebbling number \(f(G)\) of a graph \(G\) is the smallest number \(k\) such that, however \(n\) pebbles are placed on the vertices of \(G\), we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs \(G\) and \(H\), \(f(G \times H) \leq f(G)f(H)\), where \(G \times H\) represents the Cartesian product of \(G\) and \(H\). In this paper, we prove that \(f(G \times H) \leq f(G)f(H)\) when \(G\) has the two-pebbling property and \(H = K_{r,s}^{ – k}\), a graph obtained from the \(r \times s\) complete bipartite graph \(K_{r,s}\) by deleting \(k\) edges which form a matching. We also show that Graham’s conjecture holds for \(K_{r,s}^{-k_1} \times K_{m,n}^{-k_2}\).

The Hosoya polynomial of a graph \(G\) is defined as \(H(G,x) = \sum\limits_{k\geq 0} d(G,k)x^k,\)
where \(d(G, k)\) is the number of vertex pairs at distance \(k\) in \(G\). The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zheng et al. A capped zig-zag nanotube \(T(p, q)[C, D; a]\) consists of a zig-zag open-ended nanotube \(T(p, q)\) and two caps \(C\) and \(D\) with the relative position \(a\) between \(C\) and \(D\). In this paper, we give a general formula for calculating the Hosoya polynomial of any capped zig-zag nanotube. By the formula, the Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotubes \(T(p, q)[C, D; a_1]\), \(T(p, q’)[C, D; a_2]\) with \(q’ \geq q^* \geq p+1\) have the same Hosoya polynomial, where \(q^*\) is an integer that depends on the structures of \(C\) and \(D\).

Ewa Wojcicka (Journal of Graph Theory, \(14(1990), 205-215)\) showed that every connected, 3-color-critical graph on more than 6 vertices has a Hamiltonian path. Henning et al. (Discrete Mathematics, \(161(1996), 175-184)\) defined a graph \(G\) to be \(k\)-\((\gamma, d)\)-critical graph if \(\gamma(G) = k\) and \(\gamma(G + uv) = k – 1\) for each pair \(u, v\) of nonadjacent vertices of \(G\) that are at distance at most \(d\) apart. They asked if a 3-\((\gamma, 2)\)-critical graph must contain a dominating path. In this paper, we show that every connected, 3-\((\gamma, 2)\)-critical graph must contain a dominating path. Further, we show that every connected, 3-\((\gamma, 2)\)-critical graph on more than 6 vertices has a Hamiltonian path.

Let \(d(u,v)\) denote the distance between two distinct vertices of a connected graph \(G\) and \(diam(G)\) be the diameter of \(G\). A radio labeling \(f\) of \(G\) is an assignment of positive integers to the vertices of \(G\) satisfying \(d(u,v) + |f(u) – f(v)| \geq diam(G) + 1\). The maximum integer in the range of the labeling is its span. The radio number of \(G\), denoted by \(rn(G)\), is the minimum possible span. In \([7]\) M. Farooq et al. found the lower bound for the radio number of generalized gear graph. In this paper, we give an upper bound for the radio number of generalized gear graph, which coincides with the lower bound found in \([7]\).

In this paper, we study codes over polynomial rings and establish a connection to Jacobi Hilbert modular forms, specifically Hilbert modular forms over the totally real field via the complete weight enumerators of codes over polynomial rings.