For \(n \geq 1\), we let \(a_n\) count the number of nonempty subsets \(S\) of \(\{1,2,3,\ldots,n\} = [n]\), where the size of \(S\) equals the minimal element of \(S\). Such a subset is called an extraordinary subset of \([n]\), and we find that \(a_n = F_n\), the \(n\)th Fibonacci number. Then, for \(n \geq k \geq 1\), we let \(a(n, k)\) count the number of times the integer \(k\) appears among these \(a_n\) extraordinary subsets of \(n\). Here we have \(a(n, k) = a(n-1, k) + a(n-2, k-1)\), for \(n \geq 3\) and \(n > k \geq 2\). Formulas and properties for \(t_n = \sum_{k=1}^n a(n, k)\) and \(s_n = \sum_{k=1}^n ka(n, k)\) are given for \(n \geq 1\). Finally, for fixed \(n \geq 1\), we find that the sequence \(a(n, k)\) is unimodal and examine the maximum element for the sequence. In this context, the Catalan numbers make an entrance.

The cycle length distribution (CLD) of a graph of order \(n\) is \((c_1, c_2, \ldots, c_n)\), where \(c_i\) is the number of cycles of length \(i\), for \(i = 1, 2, \ldots, n\). For an integer sequence \((a_1, a_2, \ldots, a_n)\), we consider the problem of characterizing those graphs \(G\) with the minimum possible edge number and with \(\text{CLD}(G) = (c_1, c_2, \ldots, c_n)\) such that \(c_i \geq a_i\) for \(i = 1, 2, \ldots, n\). The number of edges in such a graph is denoted by \(g(a_1, a_2, \ldots, a_n)\). In this paper, we give the lower and upper bounds of \(g(0, 0, k, \ldots, k)\) for \(k = 2, 3, 4\).

Two-fold automorphisms (or “TF-isomorphisms”) of graphs are a generalisation of automorphisms. Suppose \(\alpha, \beta\) are two permutations of \(V = V(G)\) such that for any pair \((u,v)\), \(u, v \in V\), \((u,v)\) is an arc of \(G\) if and only if \((\alpha(u), \beta(v))\) is an arc of \(G\). Such a pair of permutations is called a two-fold automorphism of \(G\). These pairs form a group that is called the two-fold automorphism group. Clearly, it contains all the pairs \((\alpha, \alpha)\) where \(\alpha\) is an automorphism of \(G\). The two-fold automorphism group of \(G\) can be larger than \(\text{Aut}(G)\) since it may contain pairs \((\alpha, \beta)\) with \(\alpha \neq \beta\). It is known that when this happens, \(\text{Aut}(G) \times \mathbb{Z}_2\) is strictly contained in \(\text{Aut}(G \times K_2)\). In the literature, when this inclusion is strict, the graph \(G\) is called unstable.

Now let \(\Gamma \leq S_V \times S_V\). A two-fold orbital (or “TF-orbital”) of \(F\) is an orbit of the action \((\alpha, \beta) : (u,v) \mapsto (\alpha(u), \beta(v))\) for \((\alpha, \beta) \in \Gamma\) and \(u,v \in V\). Clearly, \(\Gamma\) is a subgroup of the TF-automorphism group of any of its TF-orbitals. We give a short proof of a characterization of TF-orbitals which are disconnected graphs and prove that a similar characterization of TF-orbitals which are digraphs might not be possible. We shall also show that the TF-rank of \(\Gamma\), that is the number of its TF-orbitals, can be equal to \(1\) and we shall obtain necessary and sufficient conditions on I for this to happen.

We define a new fairness notion on edge-colorings, requiring that the number of vertices in the subgraphs induced by the edges of each color are within one of each other. Given a (not necessarily proper) \( k \)-edge-coloring of a graph \( G \), for each color \( i \in \mathbb{Z}_k \), let \( G[i] \) denote the (not necessarily spanning) subgraph of \( G \) induced by the edges colored \( i \). Let \( \nu_{i}(G) = |V(G[i])| \). Formally, a \( k \)-edge-coloring of a graph \( G \) is said to be vertex-equalized if for each pair of colors \( i, j \in \mathbb{Z}_k \), \( |\nu_{i}(G) – \nu_{j}(G)| \leq 1 \). In this paper, a characterization is found for connected graphs that have vertex-equalized \( k \)-edge-colorings for each \( k \in \{2, 3\} \) (see Corollary 4.1 and Corollary 4.2).

Let \( G = (V, E) \) be a graph. The open neighborhood of a vertex \( v \in V \) is the set \( N(v) = \{u \mid uv \in E\} \) and the closed neighborhood of \( v \) is the set \( N[v] = N(v) \cup \{v\} \). The open neighborhood of a set \( S \) of vertices is the set \( N(S) = \bigcup_{v \in S} N(v) \), while the closed neighborhood of a set \( S \) is the set \( N[S] = \bigcup_{v \in S} N[v] \). A set \( S \subset V \) dominates a set \( T \subset V \) if \( T \subseteq N[S] \), written \( S \rightarrow T \). A set \( S \subset V \) is a dominating set if \( N[S] = V \); and is a minimal dominating set if it is a dominating set, but no proper subset of \( S \) is also a dominating set; and is a \( \gamma \)-set if it is a dominating set of minimum cardinality. In this paper, we consider the family \( \mathcal{D} \) of all dominating sets of a graph \( G \), the family \( \mathcal{MD} \) of all minimal dominating sets of a graph \( G \), and the family \( \Gamma \) of all \( \gamma \)-sets of a graph \( G \). The study of these three families of sets provides new characterizations of the distance-2 domination number, the upper domination number, and the upper irredundance number in graphs.

Irregular-spotty-byte error control codes were devised by the author in [2] and their properties were further studied in [3] and [4]. These codes are suitable for semi-conductor memories where an I/O word is divided into irregular bytes not necessarily of the same length. The \(i\)-spotty-byte errors are defined as \(t_i\) or fewer bit errors in an \(i\)-byte of length \(n_i\), where \(1 \leq t_i \leq n_i\) and \(1 \leq i \leq s\). However, an important and practical situation is when \(i\)-spotty-byte errors caused by the hit of high energetic particles are confined to \(i\)-bytes of the same size only which are aligned together or in words errors occur usually in adjacent RAM chips at a particular time. Keeping this view, in this paper, we propose a new model of \(i\)-spotty-byte errors, viz. uniform \(i\)-spotty-byte errors and present a new class of codes, viz. uniform \(i\)-spotty-byte error control codes which are capable of correcting all uniform \(i\)-spotty-byte errors of \(i\)-spotty measure \( \mu \) (or less). The study made in this paper will be helpful in designing modified semi-conductor memories consisting of irregular RAM chips with those of equal length aligned together.

Let \( \mathcal{M} \) denote the set of matrices over some semiring. An upper ideal of matrices in \( \mathcal{M} \) is a set \( \mathcal{U} \) such that if \( A \in \mathcal{U} \) and \( B \) is any matrix in \( \mathcal{M} \), then \( A + B \in \mathcal{U} \). We investigate linear operators that strongly preserve certain upper ideals (that is, linear operators on \( \mathcal{M} \) with the property that \( X \in \mathcal{U} \) if and only if \( T(X) \in \mathcal{U} \)). We then characterize linear operators that strongly preserve sets of tournament matrices and sets of primitive matrices. Specifically, we show that if \( T \) strongly preserves the set of regular tournaments when \( n \) is odd or nearly regular tournaments when \( n \) is even, then for some permutation matrix \( P \), \( T(X) = P^{t}XP \) for all matrices \( X \) with zero main diagonal, or \( T(X) = P^{t}X^{t}P \) for all matrices \( X \) with zero main diagonal. Similar results are shown for linear operators that strongly preserve the set of primitive matrices whose exponent is \( k \) for some values of \( k \), and for those that strongly preserve the set of nearly reducible primitive matrices.

For a poset \( P = (V(P), \leq_P) \), the strict semibound graph of \( P \) is the graph \( ssb(P) \) on \( V(ssb(P)) = V(P) \) for which vertices \( u \) and \( v \) of \( ssb(P) \) are adjacent if and only if \( u \neq v \) and there exists an element \( x \in V(P) \) distinct from \( u \) and \( v \) such that \( x \leq_P u,v \) or \( u,v \leq_P x \). We prove that a poset \( P \) is connected if and only if the induced subgraph \(\langle max(P)\rangle_{ssb(P)}\) is connected. We also characterize posets whose strict semibound graphs are triangle-free.

In this paper, we revisit LE graphs, find the minimum \( \lambda \) for decomposition of \( \lambda K_n \) into these graphs, and show that for all viable values of \( \lambda \), the necessary conditions are sufficient for LE-decompositions using cyclic decompositions from base graphs.

We determine the signless Laplacian spectrum for the \( H \)-join of regular graphs \( G_1, \ldots, G_p \). We also find an expression and upper bounds for the signless Laplacian spread of the \( H \)-join of regular graphs \( G_1, \ldots, G_p \).