In this paper we define new generalizations of Fibonacci numbers and Lucas numbers in the distance sense. These generalizations are closely related to the concept of \((2,k )\)-distance Fibonacci numbers presented in \([10]\). We show some applications of these numbers in number decompositions and we also define a new type of Lucas numbers.

For a vector \({R} = (r_1, r_2, \ldots, r_m)\) of non-negative integers, a mixed hypergraph \(\mathcal{H}\) is a realization of \({R}\) if its chromatic spectrum is \({R}\). In this paper, we determine the minimum number of vertices of realizations of a special kind of vectors \({R}_2\). As a result, we partially solve an open problem proposed by Král in \(2004\).

A strong edge-coloring is a proper edge-coloring such that two edges with the same color are not allowed to lie on a path of length three. The strong chromatic index of a graph \(G\), denoted by \(s'(G)\), is the minimum number of colors in a strong edge-coloring. We denote the degree of a vertex \(v\) by \(d(v)\). Let the \({Ore-degree}\) of a graph \(G\) be the maximum value of \(d(u) + d(v)\), where \(u\) and \(v\) are adjacent vertices in \(G\). Let \(F_3\) denote the graph obtained from a \(5\)-cycle by adding a new vertex and joining it to a pair of nonadjacent vertices of the \(5\)-cycle. In \(2008\), Wu and Lin [J. Wu and W. Lin, The strong chromatic index
of a class of graphs, Discrete Math., \(308 (2008), 6254-6261]\) studied the strong chromatic index with respect to the Ore-degree. Their main result states that if a connected graph \(G\) is not \(F_3\) and its Ore-degree is \(5\), then \(s'(G) \leq 6\). Inspired by the result of Wu and Lin, we investigate the strong edge-coloring of graphs with Ore-degree 6. We show that each graph \(G\) with Ore-degree \(6\) has \(s'(G) \leq 10\). With the further condition that \(G\) is bipartite, we have \(s'(G) \leq 9\). Our results give general forms of previous results about strong chromatic indices of graphs with maximum degree \(3\).

For a graph \(G\), an edge labeling of \(G\) is a bijection \(f: E(G) \to \{1, 2, \ldots, |E(G)|\}\). The \emph{induced vertex sum} \(f^*\) of \(f\) is a function defined on \(V(G)\) given by \(f^+(u) = \sum_{uv \in E(G)} f(uv)\) for all \(u \in V(G)\). A graph \(G\) is called \emph{antimagic} if there exists an edge labeling of \(G\) such that the induced vertex sum of the edge labeling is injective. Hartsfield and Ringel conjectured in 1990 that all connected graphs except \(K_2\) are antimagic. A spider is a connected graph with exactly one vertex of degree exceeding \(2\). This paper shows that all spiders are antimagic.

In this paper, we consider the problem of determining precisely which graphic matroids \(M\) have the property that the splitting operation,by every pair of elements, on \(M\) yields a cographic matroid. This problem is solved by proving that there are exactly three minorminimal graphs that do not have this property.

In this paper, we give a new and interesting identities of Boole and Euler polynomials which are derived from the symmetry properties of the \(p\)-adic fermionic integrals on \(\mathbb{Z}_p\).

In this paper we address the problem of construction of critical sets
in \(F\)-squares of the form \(F(2n; 2, 2,……… ,2)\). We point out that the
critical set in \(F(2n; 2,2, ……… ,2)\) obtained by Fitina, Seberry and
Sarvate \((1999)\) is not correct and prove that in the given context a
proper subset is a critical set.

A connected graph \(G = (V(G), E(G))\) is called a quasi-tree graph if there exists a vertex \(u_0 \in V(G)\) such that \(G – u_0\) is a tree. Let \(\mathcal{P}(2k) := \{G: G \text{ is a quasi-tree graph on } 2k \text{ vertices with perfect matching}\}\), and \(\mathcal{P}(2k, d_0) := \{G: G \in \mathcal{P}(2k), \text{ and there is a vertex } u_0 \in V(G) \text{ such that } G – u_0 \text{ is a tree with } d_G(u_0) = d_0\}\). In this paper, the maximal indices of all graphs in the sets \(\mathcal{P}(2k)\) and \(\mathcal{P}(2k, d_0)\) are determined, respectively. The corresponding extremal graphs are also characterized.

A combinatorial sum for the Stirling numbers of the second kind is generalized. This generalization provides a new explicit formula for the binomial sum \(\sum_{k=0}^{n}k^ra^kb^{n-k} \binom{n}{k}\), where \(a, b \in \mathbb{R} – \{0\}\) and \(n, r \in \mathbb{N}\). As relevant special cases, simple explicit expressions for both the binomial sum \(\sum_{k=0}^{n} k^r\binom{n}{k} \) and the raw moment of order \(r\) of the binomial distribution \(B(n, p)\) are given. All these sums are expressed in terms of generalized \(r\)-permutations.

Let \(G\) be a simple connected graph with vertex set \(V(G)\). The Gutman index \(\text{Gut}(G)\) of \(G\) is defined as \(\text{Gut}(G) = \sum\limits_{\{x,y\} \subseteq V(G)} d_G(x) d_G(y) d_G(x,y)\), where \(d_G(x)\) is the degree of vertex \(v\) in \(G\) and \(d_G(x,y)\) is the distance between vertices \(x\) and \(y\) in \(G\). In this paper, the second-minimum Gutman index of unicyclic graphs on \(n\) vertices and girth \(m\) is characterized.