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.

We prove a two-point concentration for the tree domination number of the random graph \(G_{n,p}\) provided \(p\) is constant or \(p \to 0\) sufficiently slow.

A 2-independent set in a graph \(G\) is a subset \(J\) of the vertices such that the distance between any two vertices of \(J\) in \(G\) is at least three. The number of 2-independent sets of a graph \(G\) is denoted by \(i_2(G)\). For a forest \(F\), \(i_2(F – e) > i_2(F)\) for each edge \(e\) of \(F\). Hence, we exclude all forests having isolated vertices. A forest is said to be extra-free if it has no isolated vertex. In this paper, we determine the \(k\)-th largest number of 2-independent sets among all extra-free forests of order \(n \geq 2\), where \(k = 1, 2, 3\). Extremal graphs achieving these values are also given.