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.

The clique-chromatic number of a graph is the least number of colors on the vertices of the graph without a monocolored maximal clique of size at least two.In \(2004\), Bacsé et al. proved that the family of line graphs has no bounded clique-chromatic number. In particular, the Ramsey numbers provide a sequence of the line graphs of complete graphs with no bounded clique-chromatie number. We
complete this result by giving the exact values of the clique-chromatic numbers of the line graphs of complete graphs in terms of Ramsey numbers. Furthermore, the clique-chromatic numbers of the line graphs of triangle-free graphs are characterized.

The current article focuses on the generalized \(k\)-Pell \((p, i)\)-numbers for \(k = 1, 2, \ldots\) and \(0 \leq i \leq p\). It introduces the generalized \(k\)-Pell \((p, i)\)-numbers and their generating matrices and generating functions. Some interesting identities are established.

For a graph \(G\), let \({Z}(G)\) be the total number of matchings in \(G\). For a nontrivial graph \(G\) of order \(n\) with vertex set \(V(G) = \{v_1, \ldots, v_n\}\), Cvetković et al. \([2]\) defined the triangle graph of \(G\), denoted by \(R(G)\), to be the graph obtained by introducing a new vertex \(v_{ij}\) and connecting \(u_{ij}\) both to \(v_i\) and to \(v_j\) for each edge \(v_iv_j\) in \(G\). In this short paper, we prove that for a connected graph \(G\), if \({Z}(R(G)) \geq (\frac{1}{2}n-\frac{1}{2}+\frac{5}{2n})^2\), then \(G\) is traceable. Moreover, for a connected graph \(G\), we give sharp upper bounds for \({Z}(R(G))\) in terms of clique number, vertex connectivity, and spectral radius of \(G\), respectively.