The main aim of this paper is to construct an extension of Appell’s hypergeometric functions by means of modified Beta functions \(B(x, y; p)\). We give integral representations for these functions and obtain some relations for these functions and extended Gauss hypergeometric function via decomposition operators defined by Burchnall and Chaundy. Furthermore, we present some transformation formulas for the first and second kind of extended Appell’s hypergeometric functions. Also, we give some relations between the first kind of extended Appell’s hypergeometric functions, Whittaker, and Modified Bessel functions.

Informally, a \(\epsilon\)-switchable \(G\)-design is a decomposition of the complete graph into subgraphs of isomorphic copies of \(G\) which have the property that they remain a \(G\)-decomposition when \(\epsilon\)-edge switches are made to the subgraphs. This paper determines the spectrum of \(\epsilon\)-switchable \(G\)-designs where \(G\) is a kite (a triangle with an edge attached) and \(\epsilon\) takes \(t\)-edge, \(h\)-edge, and \(l\)-edge.

In this paper, we use a simple method to derive different recurrence relations on the Tribonacci numbers and their sums. By using the companion matrices and generating matrices, we obtain more identities on the Tribonacci numbers and their sums, which are more general than those given in the literature [E. Kilic, Tribonacci Sequences with Certain Indices and Their Sum, Ats Combinatoria \(86 (2008),13-22]\).

A \((2,1)\)-total labeling of a graph \(G\) is a labeling of vertices and edges, such that:(1) any two adjacent vertices of \(G\) receive distinct integers,(2) any two adjacent edges receive distinct integers, and (3) a vertex and its incident edges receive integers that differ by at least 2 in absolute value.The span of a \((2,1)\)-total labeling is the difference between the maximum label and the minimum label.We note the minimum span \(\lambda_2^T(G)\).In this paper, we prove that if \(G\) is a planar graph with \(\Delta \leq 3\) and girth \(g \geq 18\), then \(\lambda_2^T(G) \leq 5\). If \(G\) is a planar graph with \(\Delta \leq 4\) and girth \(g \geq 12\), then \(\lambda_2^T(G) \leq 7\).

If \(X\) is a geodesic metric space and \(x_1, x_2, x_3 \in X\), a geodesic triangle \(T = \{x_1, x_2, x_3\}\) is the union of the three geodesics \([x_1 x_2], [x_2 x_3]\) and \([x_3 x_1]\) in \(X\). The space \(X\) is \(\delta\)-hyperbolic (in the Gromov sense) if any side of \(T\) is contained in a \(\delta\)-neighborhood of the union of the two other sides, for every geodesic triangle \(T\) in \(X\). We denote by \(\delta(X)\) the sharp hyperbolicity constant of \(X\), i.e. \(\delta(X) := \inf\{\delta \geq 0: X \text{ is } \delta\text{-hyperbolic}\}\). In this paper, we find some relations between the hyperbolicity constant of a graph and its order, girth, cycles, and edges. In particular, if \(g\) denotes the girth, we prove \(\delta(G) \geq g(G)/4\) for every (finite or infinite) graph; if \(G\) is a graph of order \(n\) and edges with length \(k\) (possibly with loops and multiple edges), then \(\delta(G) \leq nk/4\). We find a large family of graphs for which the first (non-strict) inequality is in fact an equality; besides, we characterize the set of graphs with \(\delta(G) = nk/4\). Furthermore, we characterize the graphs with edges of length \(k\) with \(\delta(G) < k\).

A proper edge coloring \(c\) of a graph \(G\) is said to be acyclic if \(G\) has no bicolored cycle with respect to \(c\). It is proved that every triangle-free toroidal graph \(G\) admits an acyclic edge coloring with \((\Delta(G) + 5)\) colors. This generalizes a theorem from \([8]\).

Let \(\mathcal{J}_n\) be the set of tricyclic graphs of order \(n\). In this paper, we use a new proof to determine the unique graph with maximal spectral radius among all graphs in \(\mathcal{J}_n\) for each \(n \geq 4\). Also, we determine the unique graph with minimal least eigenvalue among all graphs in this class for each \(n \geq 52\). We can observe that the graph with maximal spectral radius is not the same as the one with minimal least eigenvalue in \(\mathcal{J}_n\), which is different from those on the unicyclic and bicyclic graphs.

Let \(G\) be a connected simple graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \sum_{u,v \in V(G)} (d(u, v) + d^2(u,v)),\) with the summation going over all pairs of vertices in \(G\). In this paper, we determine the extremal unicyclic graphs with given matching number and minimal hyper-Wiener index.

Robertson \(([5])\) and independently, Bondy \(([1])\) proved that the generalized Petersen graph \(P(n, 2)\) is non-hamiltonian if \(n \equiv 5 \pmod{6}\), while Thomason \([7]\) proved that it has precisely \(3\) hamiltonian cycles if \(n \equiv 3 \pmod{6}\). The hamiltonian cycles in the remaining generalized Petersen graphs were enumerated by Schwenk \([6]\). In this note we give a short unified proof of these results using Grinberg’s theorem.

We present some binomial identities for sums of the bivariate Fibonacci polynomials and for weighted sums of the usual Fibonacci polynomials with indices in arithmetic progression.