In this paper, we present two criteria for a sequence lying along a ray of a combinatorial triangle to be unimodal, and give a correct
proof for the result of Belbachir and Szalay on unimodal rays of the generalized Pascal’s triangle.

In this paper, we introduce the notion of derivation in lattice implication algebra, and consider the properties of derivations in lattice implication algebras. We give an equivalent condition to be a derivation of a lattice implication algebra. Also, we characterize the fixed set \(Fix_d(L)\) and \(Kerd\) by derivations. Moreover, we prove that if \(d\) is a derivation of a lattice implication algebra, every filter \(F\) is \(d\)-invariant.

In this paper, we derive a family of identities on the arbitrary subscripted Fibonacci and Lucas numbers. Furthermore, we construct the tridiagonal and symmetric tridiagonal family of matrices whose determinants form any linear subsequence of the Fibonacci numbers and Lucas numbers. Thus, we give a generalization of the results presented in Nalli and Civciv [A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos, Solitons and Fractals \(2009;40(1):355 .61]\) and Cahill and Narayan [N. D. Cahill, D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, The Fibonacci Quarterly, \(2004;42(1):216–221]\).

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we determine the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.

For a poset \(P = (X, \leq_ P)\), the strict-double-bound graph (\(sDB\)-graph \(sDB(P)\)) is the graph on \(X\) for which vertices \(u\) and \(v\) of \(sDB(P)\) are adjacent if and only if \(u \neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) such that \(x \leq_ P y\) and \(x \leq_P v \leq_P y\). The strict-double-bound number \(\zeta(G)\) of a graph \(G\) is defined as \(\min\{n; G \cup \overline{K}_n \text{ is a strict-double-bound graph}\}\).

We obtain that for a spider \(S_{n,m}\) (\(n,m > 3\)) and a ladder \(L_n\) (\(n \geq 4\)), \(\left\lceil2\sqrt{nm}\right\rceil \leq \zeta(S_{n,m}) \leq n+m\), \(\zeta(S_{n,n}) = 2n\), and \(\left\lceil 2\sqrt{3n+2}\right\rceil \leq \zeta(L_n) \leq 2n\).

We conjectured in \([3]\) that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the \(3\)-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete \(k\)-partite, and matrogenic cyclic graphs are weakly split.

Let \((X,\mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition and let \(G_i\), \(i = 1,\ldots,\mu\), be nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i | B \in \mathcal{B}\}\), where \(\mathcal{B_i}\) is a subgraph of \(B\)  isomorphic to \(G_i\). A \(\{G_1,G_2,\ldots,G_\mu\}\)-metamorphosis of \((X,\mathcal{B})\) is a rearrangement, for each \(i=1,\ldots,\mu\), of the edges of \(\bigcup_{B\in B}(E(B)\setminus\mathcal{B}_i))\) into a family \(\mathcal{F}_i\) of copies of \(G_i\) with a leave \(L_i\), such that \((X,\mathcal{B}_i \cup \mathcal{F}_i,L_i)\) is a maximum packing of \(\lambda H\) with copies of \(G_i\). In this paper, we give a complete answer to the existence problem of an \(S_\lambda(2,4,7)\) having a \(\{C_4, K_3 + e\}\)-metamorphosis.

For a positive integer \(m\), where \(1 \leq m \leq n\), the \(m\)-competition index (generalized competition index) of a primitive digraph \(D\) of order \(n\) is the smallest positive integer \(k\) such that for every pair of vertices \(x\) and \(y\), there exist \(m\) distinct vertices \(v_1, v_2, \ldots, v_m\) such that there exist walks of length \(k\) from \(x\) to \(v_i\) and from \(y\) to \(v_i\) for \(1 \leq i \leq m\). In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.

We give a new combinatorial bijection between a certain set of balanced modular tableaux of Gusein-Zade, Luengo, and Melle-Hernandez and \(k\)-ribbon shapes. In addition, we also use the Schensted algorithm for the rim hook tableaux of Stanton and White to write down an explicit generating function for these balanced modular tableaux.

A \((k;g)\)-cage is a graph with the minimum order among all \(k\)-regular graphs with girth \(g\). As a special family of graphs, \((k;g)\)-cages have a number of interesting properties. In this paper, we investigate various properties of cages, e.g., connectivity, the density of shortest cycles, bricks and braces.