It is widely recognized that certain graph-theoretic extremal questions play a major role in the study of communication network vulnerability. These extremal problems are special cases of questions concerning the realizability of graph invariants. We define a CS(\(p, q, \lambda, \delta\)) graph as a connected, separable graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). In this notation, if the “CS” is omitted the graph is not necessarily connected and separable. An arbitrary quadruple of integers \((a, b, c, d)\) is called CS(\(p, q, A, 5\)) realizable if there is a CS(\(p, q, \lambda, \delta\)) graph with \(p = a, q = b, \lambda = c\) and \(\delta= d\). Necessary and sufficient conditions for a quadruple to be CS(\(p, q,\lambda, \delta\)) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta), (p, q, \lambda, \Delta), (p, q, \delta, \Delta), (p, q, \lambda, \delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(A\) denotes the maximum degree for all points in a graph and \(\lambda\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa), (p, q, \lambda), (p, q, \delta), (p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.

We use \(k\)-trees to generalize the sequence of Motzkin numbers and show that Baxter’s generalization of Temperley-Lieb operators is a special case of our generalization of Motzkin numbers. We also obtain a recursive summation formula for the terms of \(3\)-Motzkin numbers and investigate some asymptotic properties of the terms of \(k\)-Motzkin numbers.

In this article, defining the matrix extensions of the Fibonacci and Lucas numbers, we start a new approach to derive formulas for some integer numbers which have appeared, often surprisingly, as answers to intricate problems, in conventional and in recreational Mathematics. Our approach provides a new way of looking at integer sequences from the perspective of matrix algebra, showing how several of these integer sequences relate to each other.

For a finite group \(G\) the commutativity degree,

\[d(G)=\frac{|\{(x,y)|x,y \in G, xy=yx\}|}{|G|^2}\]

is defined and studied by several authors and when \(d(G) \geq \frac{1}{2}\) it is proved by P. Lescot in 1995 that \(G\) is abelian , or \(\frac{G}{Z(G)}\) is elementary abelian with \(|G’| = 2\), or \(G\) is isoclinic with \(S_3\) and \(d(G) = 1\). The case when \(d(G) < \frac{1}{2}\) is of interest to study. In this paper we study certain infinite classes of finite groups and give explicit formulas for \(d(G)\). In some cases the groups satisfy \(\frac{1}{4} < d(G) < \frac{1}{2}\). Some of the groups under study are nilpotent of high nilpotency classes.

In this paper, we construct a new infinite family of balanced binary sequences of length \(N = 4p\), \(p \equiv 5 \pmod{8}\) with optimal autocorrelation magnitude \(\{N, 0, \pm 4\}\).

The cocircuits of a splitting matroid \(M_{i,j}\) are described in terms of the cocircuits of the original matroid \(M\).

Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called an \(f\)-factor if \(d_F(x) = f(x)\) for every \(x \in V(F)\). In this paper, we present some sufficient conditions for the existence of \(f\)-factors and connected \((f-2, f)\)-factors in \(K_{1,n}\)-free graphs. The conditions involve the minimum degree, the stability number, and the connectivity of graph \(G\).

We classify the minimal blocking sets of size 15 in \(\mathrm{PG}(2,9)\). We show that the only examples are the projective triangle and the sporadic example arising from the secants to the unique complete 6-arc in \(\mathrm{PG}(2,9)\). This classification was used to solve the open problem of the existence of maximal partial spreads of size 76 in \(\mathrm{PG}(3,9)\). No such maximal partial spreads exist \([13]\). In \([14]\), also the non-existence of maximal partial spreads of size 75 in \(\mathrm{PG}(3,9)\) has been proven. So, the result presented here contributes to the proof that the largest maximal partial spreads in \(\mathrm{PG}(3,q=9)\) have size \(q^2-q+2=74\).

Our work in this paper is concerned with a new kind of fuzzy ideal of a \(K\)-algebra called an \((\in, \in \vee_q)\)-fuzzy ideal. We investigate some interesting properties of \((\in, \in \vee_q)\)-fuzzy ideals of \(K\)-algebras. We study fuzzy ideals with thresholds which is a generalization of both fuzzy ideals and \((\in, \in \vee_q)\)-fuzzy ideals. We also present characterization theorems of implication-based fuzzy ideals.

Let \(G\) be a digraph. For two vertices \(u\) and \(v\) in \(G\), the distance \(d(u,v)\) from \(u\) to \(v\) in \(G\) is the length of the shortest directed path from \(u\) to \(v\). The eccentricity \(e(v)\) of \(v\) is the maximum distance of \(v\) to any other vertex of \(G\). A vertex \(u\) is an eccentric vertex of \(v\) if the distance from \(v\) to \(u\) is equal to the eccentricity of \(v\). The eccentric digraph \(ED(G)\) of \(G\) is the digraph that has the same vertex set as \(G\) and the arc set defined by: there is an arc from \(u\) to \(v\) if and only if \(v\) is an eccentric vertex of \(u\). In this paper, we determine the eccentric digraphs of digraphs for various families of digraphs and we get some new results on the eccentric digraphs of the digraphs.