The aim of our paper is to introduce generalized neighborhood bases and \(gn-T_2\)-spaces. \((\psi, \psi’)\)-continuity, sequentially \((\psi, \psi’)\)-continuity, and \(\psi\)-convergency are investigated on strong generalized first countable spaces, and also two results about \(\psi\)-convergency on \((\psi, \psi’)\)-\(T_2\)-spaces are given.

For a graph \(H\) and an integer \(k \geq 2\), let \(\sigma_k(H)\) denote the minimum degree sum of \(k\) independent vertices of \(H\). We prove that if a connected claw-free graph \(G\) satisfies \(\sigma_{k+1}(G) \geq |G| – k\), then \(G\) has a spanning tree with at most \(k\) leaves. We also show that the bound \(|G| – k\) is sharp and discuss the maximum degree of the required spanning trees.

Define the conditional recurrence sequence \(q_n = aq_{n-1} + bq_{n-2}\) if \(n\) is even, \(q_n = bq_{n-1} + cq_{n-2}\) if \(n\) is odd, where \(q_0 = 0, q_1 = 1\). Then \(q_n\) satisfies a fourth-order recurrence while both \(q_{2n}\) and \(q_{2n+1}\) satisfy a second-order recurrence.

Analogously to a Lucas pseudoprime, we define a composite number \(n\) to be a conditional Lucas pseudoprime (clpsp) if \(n\) divides \(q_{n – (\frac{\Delta}{n})}\), where \(\Delta = a^2 + b^2 + 4ab\) and \((\frac{\Delta}{n})\) denotes the Jacobi symbol. We prove that if \((n, 2ab\Delta) = 1\), then there are infinitely many conditional Lucas pseudoprimes. We also address the question, given an odd composite integer \(n\), for how many pairs \((a, b)\) is \(n\) a conditional Lucas pseudoprime?

Let \(G\) be a simple connected graph with \(n\) vertices. Denoted by \(L(G)\) the Laplacian matrix of G. In this paper, we present a sequence of graphs \({G_n}\) with \(\lim\limits_{n\to \infty} \mu_3(G_n) = 1.5550\) by investigating the eigenvalues of the line graphs of \({G_n}\). Moreover, we prove that the limit is the minimal limit point of the third largest Laplacian eigenvalues of graphs.

Two cycles are said to be intersecting if they share at least one common vertex. Let \(\chi'(G)\) and \(\chi”(G)\) denote the list edge chromatic number and list total chromatic number of a graph \(G\), respectively.In this paper, we proved that for any toroidal graph G without intersecting triangles, \(\chi'(G) \leq \Delta(G) +1\) and \(\chi”(G) \leq \Delta(G)+2\) if \(\Delta(G) \geq 6\), and \(\chi'(G) = \Delta(G)\) if \(\Delta(G) \geq 8\).

Graphs which are derived from the same graph are called homeomorphic graphs or simply homeomorphs. A \(K_4\)-homeomorph denoted by
\(K_4(a,,c,d,e, f)\) is obtained by subdividing the six paths of a complete graph with four vertices into \(a, b, c,d, e, f\) number of segments, respectively.In this paper, we shall study the chromaticity of \(K_4(a, b,c,d,e, f)\) with exactly two non-adjacent paths of length two. We also give a sufficient and necessary condition for all the graphs in this family to be chromatically
unique.

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a
non-negative integer (label) such that for any two vertices \(u\) and \(v\), \(|f(u) — f(v)| \geq d — d(u,v)\), where \(d(u, v)\)
is the distance between \(u\) and \(v\). The span of an antipodal labeling f is \(\max{f(u) — f(v) : u,v \in V(G)}\). The
antipodal number for G, denoted by an\((G)\), is the minimum span of an antipodal labeling for \(G\). Let \(C_n\) denote
the cycle on n vertices. Chartrand \(et al\). \([4]\) determined the value of an\((C_n)\) for \(n \equiv 2 \pmod 4\). In this article we
obtain the value of an\((C_n)\) for \(n \equiv 1 \pmod 4\), confirming a conjecture in \([4]\). Moreover, we settle the case \(n \equiv 3 \pmod 4\), and improve the known lower bound and give an upper bound for the case \(n \equiv 0 \pmod 4\).

We classify all embeddings \(\theta\) : \(PG(n,\mathbb{K}) \rightarrow PG(d,\mathbb{F})\), with \(d \geq \frac{n(n+1)}{2}\)
and \(\mathbb{K},\mathbb{F}\) skew fields with \(|\mathbb{K}| > 2\), such that \(\theta\) maps the set of points of each line of \(PG(n, \mathbb{K})\) to a set of coplanar points of \(PG(n, \mathbb{F})\), and such that the image of \(\theta\) generates \(PG(d, \mathbb{F})\). It turns out that \(d = \frac{1}{2}n(n + 3)\) and all examples “essentially” arise from a similar “full” embedding \(\theta’\) : \(PG(n, \mathbb{K}) \rightarrow PG(d, \mathbb{K})\) by identifying \(\mathbb{K}\) with subfields of F and embedding \(PG(d, \mathbb{K})\) into \(PG(d, \mathbb{F})\) by several ordinary field extensions. These “full” embeddings satisfy one more property and are classified in \([5]\). They relate to the quadric Verone-sean of \(PG(n, \mathbb{K})\) in \(PG(d, \mathbb{K})\) and its projections from subspaces of \(PG(n, \mathbb{K})\) generated by sub-Veroneseans (the point sets corresponding to subspaces of \(PG(n, \mathbb{K})\), if \(\mathbb{K}\) is commutative, and to a degenerate analogue of this, if \(\mathbb{K}\) is noncommutative.

Let \(\mathbb{N}\) be the set of all positive integers, and \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\). For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_h\)-magic if there exists a labeling \(f: E \rightarrow \mathbb{Z}_h \setminus \{0\}\) such that the induced vertex labeling \(f^+: V \rightarrow \mathbb{Z}_h\), defined by \(f^+(v) = \sum_{uv \in E(v)} f(uv)\), is a constant map. The integer-magic spectrum of \(G\) is the set \(\text{JM}(G) = \{h \in \mathbb{N} \mid G \text{ is } \mathbb{Z}_h\text{-magic}\}\). A sun graph is obtained from attaching a path to each pair of adjacent vertices in an \(n\)-cycle. In this paper, we show that the integer-magic spectra of sun graphs are completely determined.

Let \(e: \mathcal{S} \rightarrow \Sigma\) be a full polarized projective embedding of a dense near polygon \(\mathcal{S}\), i.e., for every point \(p\) of \(\mathcal{S}\), the set \(H_p\) of points at non-maximal distance from \(p\) is mapped by \(e\) into a hyperplane \(\Pi_p\) of \(\Sigma\). We show that if every line of \(S\) is incident with precisely three points or if \(\mathcal{S}\) satisfies a certain property (P\(_y\)) then the map \(p \mapsto \Pi_p\) defines a full polarized embedding \(e^*\) (the so-called dual embedding of \(e\)) of \(\mathcal{S}\) into a subspace of the dual \(\Sigma^*\) of \(\Sigma\). This generalizes a result of \([6]\) where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P\(_y\)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding.