A join graph is the complete union of two arbitrary graphs. An edge cover coloring is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at least one time. The maximum number of colors needed to edge cover color \(G\) is called the edge cover chromatic index of \(G\) and denoted by \(\chi’C(G)\). It is well known that any simple graph \(G\) has the edge cover chromatic index equal to \(\delta(G)\) or \(\delta(G) – 1\), where \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’C(G) = \delta(G)\), then \(G\) is of C1-Class , otherwise \(G\) is of C2-Class . In this paper, we give some sufficient conditions for a join graph to be of C1-Class.

Let \(G = (V, E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). The Wiener index of \(G\) is defined by \(W(G) = \sum_{x,y \subseteq V} d(x,y),\) where \(d(x,y)\) is the length of the shortest path from \(x\) to \(y\). The Szeged index of \(G\) is defined by \(S_z(G) = \sum_{e =uv\in E} n_u(e|G) n_v(e|G),\) where \(n_u(e|G)\) (resp. \(n_v(e|G)\)) is the number of vertices of \(G\) closer to \(u\) (resp. \(v\)) than \(v\) (resp. \(u\)). The Padmakar-Ivan index of \(G\) is defined by \(PI(G) = \sum_{e =uv \in E} [n_{eu}(e|G) + n_{ev}(e|G)],\) where \(n_{eu}(e|G)\) (resp. \(n_{ev}(e|G)\)) is the number of edges of \(G\) closer to \(u\) (resp. \(v\)) than \(v\) (resp. \(u\)). In this paper, we will consider the graph of a certain nanostar dendrimer consisting of a chain of hexagons and find its topological indices such as the Wiener, Szeged, and \(PI\) index.

In this paper, we introduce a class of digraphs called \((l,m)\)-walk-regular digraphs, a common generalization of both weakly distance-regular digraphs \([1]\) and \(k\)-walk-regular digraphs \([3]\), and give several characterizations of them about their regularity properties that are related to distance and about the number of walks of given length between vertices at a given distance.

A graph is said to be cordial if it has a 0-1 labeling that satisfies certain properties. A wheel \(W_n\) is the graph obtained from the join of the cycle \(C_n\) (\(n \geq 3\)) and the null graph \(N_1\). In this paper, we investigate the cordiality of the join and the union of pairs of wheels and graphs consisting of a wheel and a path or a cycle.

In this paper, we show new proofs of some important formulas by means of Liu’s expansion formula. Our results include a new proof of the identity for sums of two squares, a new proof of Gauss’s identity, a new proof of Euler’s identity, and a new proof of the identity for sums of four squares.

We explicitly evaluate the generating functions for joint distributions of pairs of the permutation statistics \(\text{inv}, {maj}\), and \({ch}\) over the symmetric group when both variables are set to \(-1\). We give a combinatorial proof by means of a sign-reversing involution that specializing the variables to \(-1\) in these bimahonian generating functions gives the number of two-colored permutations up to sign.

General methods for the construction of magic squares of any order have been searched for centuries. Several `standard strategies’ have been found for this purpose, such as the `knight movement’, or the construction of bordered magic squares, which played an important role in the development of general methods.

What we try to do here is to give a general and comprehensive approach to the construction of magic borders, capable of assuming methods produced in the past as particular cases. This general approach consists of a transformation of the problem of constructing magic borders to a simpler – almost trivial – form. In the first section, we give some definitions and notation. The second section consists of the exposition and proof of our method for the different cases that appear (Theorems 1 and 2). As an application of this method, in the third section we characterize magic borders of even order, giving therefore a first general result for bordered magic squares.

Although methods for the construction of bordered magic squares have always been presented as individual successful attempts to solve the problem, we will see that a common pattern underlies the fundamental mechanisms that lead to the construction of such squares. This approach provides techniques for constructing many magic bordered squares of any order, which is a first step to construct all of them, and finally know how many bordered squares are for any order. These may be the first elements of a general theory on bordered magic squares.

The main purpose of this paper is to define a pair of Konhauser matrix polynomials and obtain some properties, such as recurrence relations and matrix differential equations, for Konhauser matrix polynomials.

Studying expressions of the form \((f(z)D)^n\), where \(D = \frac{d}{dx}\) is the derivation operator, goes back to Scherk’s Ph.D. thesis in 1823. We show that this can be extended as
\(\sum{\gamma_{p;a}}(f^{(0)})^{a(0)+1}(f^{(1)})^{a(1)}\ldots (f^{(p-1)})^{a(p-1)}D^{p-\sum_i ia(i)},\) where the summation is taken over the \(p\)-tuples \((a_0, a_1, \ldots, a_{p-1})\), satisfying \(\sum_ia(i)=p-1 + ,\sum_iia(i) < p\), \(f^{(i)} = D^if\), and \(\gamma_{p;a}\) is the number of increasing trees on the vertex set \([0, p]\) having \(a(0) + 1\) leaves and having \(a(i)\) vertices with \(i\) children for \(0 < i < p\). Thus, previously known results about increasing trees lead us to some equalities containing coefficients \(\gamma_{p;a}\). In the sequel, we consider the expansion of \({(x^kD)}^p\) and coefficients appearing there, which are called generalized Stirling numbers by physicists. Some results about these coefficients and their inverses are discussed through bijective methods. Particularly, we introduce and use the notion of \((p,k)\)-forest in these arguments.