Let \(\text{ASG}(2v,\mathbb{F}_q)\) be the \(2v\)-dimensional affine-symplectic space over the finite field \(\mathbb{F}_q\), and let \(\text{ASp}_{2v}(\mathbb{F}_q)\) be the affine-symplectic group of degree \(2v\) over \(\mathbb{F}_q\). For any two orbits \(M’\) and \(M”\) of flats under \(\text{ASp}_{2v}(\mathbb{F}_q)\), let \(\mathcal{L}’\) (resp. \(\mathcal{L}”\)) be the set of all flats which are joins (resp. intersections) of flats in \(M’\) (resp. \(M”\)) such that \(M” \subseteq L’\) (resp. \(M’ \subseteq \mathcal{L}”\)) and assume the join (resp. intersection) of the empty set of flats in \(\text{ASG}(2v,\mathbb{F}_q)\) is \(\emptyset\) (resp. \(\mathbb{F}_q^{(2v)}\)). Let \(\mathcal{L} =\mathcal{L}’ \cap \mathcal{L}”\). By ordering \(\mathcal{L}’,\mathcal{L}”, \mathcal{L}\) by ordinary or reverse inclusion, six lattices are obtained. This article discusses the relations between different lattices, and computes their characteristic polynomial.

In this paper, we calculate the number of fuzzy subgroups of a special class of non-abelian groups of order \(p^3\).

This paper addresses the problem of capturing nondominated points on non-convex Pareto frontiers, which are encountered in \(E\)-convex multi-objective optimization problems. We define a nondecreasing map \(T\) which transfers a non-convex Pareto frontier to a convex Pareto frontier. An algorithm to find a piecewise linear approximation of the nondominated set of the convex Pareto frontier is applied. Finally, the inverse map of \(T\) is used to obtain the non-convex Pareto frontier.

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 \(n \equiv 0 \pmod 4\).