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 \( G_1, \ldots, G_p \).

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path \( \pi = v_1, v_2, \ldots, v_{k+1} \) is a downhill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(v_i) \geq \deg(v_{i+1}) \). Conversely, a path \( \pi = u_1, u_2, \ldots, u_{k+1} \) is an uphill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(u_i) \leq \deg(u_{i+1}) \). The downhill domination number of a graph \( G \) is defined to be the minimum cardinality of a set \( S \) of vertices such that every vertex in \( V \) lies on a downhill path from some vertex in \( S \). The uphill domination number is defined as expected. We explore the properties of these invariants and their relationships with other invariants. We also determine a Vizing-like result for the downhill (respectively, uphill) domination numbers of Cartesian products.

Set-to-Set Broadcasting is an information distribution problem in a connected graph, \( G = (V, E) \), in which a set of vertices \( A \), called originators, distributes messages to a set of vertices \( B \) called receivers, such that by the end of the broadcasting process each receiver has received the messages of all the originators. This is done by placing a series of calls among the communication lines of the graph. Each call takes place between two adjacent vertices, which share all the messages they have. Gossiping is a special case of set-to-set broadcasting, where \( A = B = V \). We use \( F(A, B, G) \) to denote the length of the shortest sequence of calls that completes the set-to-set broadcast from a set \( A \) of originators to a set \( B \) of receivers, within a connected graph \( G \). \( F(A, B, G) \) is also called the cost of an algorithm. We present bounds on \( F(A, B, G) \) for weighted and for non-weighted graphs.

Let \( \gamma_c(G) \) denote the connected domination number of the graph \( G \). A graph \( G \) is said to be connected domination edge critical, or simply \( \gamma_c \)-critical, if \( \gamma_c(G + e) < \gamma_c(G) \) for each edge \( e \in E(\overline{G}) \). We answer a question posed by Zhao and Cao concerning \( \gamma_c \)-critical graphs with maximum diameter.

A radio labeling of a simple connected graph \( G \) is a function \( f: V(G) \to \mathbb{Z}^+ \) such that for every two distinct vertices \( u \) and \( v \) of \( G \),
$$distance(u, v) + |f(u) – f(v)| \geq 1 + diameter(G).$$
The radio number of a graph \( G \) is the smallest integer \( M \) for which there exists a labeling \( f \) with \( f(v) \leq M \) for all \( v \in V(G) \). An edge-balanced caterpillar graph is a caterpillar graph that has an edge such that removing this edge results in two components with an equal number of vertices. In this paper, we determine the radio number of particular edge-balanced caterpillars as well as improve the lower bounds of the radio number of other edge-balanced caterpillars.

It is known that an ordered \(\rho\)-labeling of a bipartite graph \( G \) with \( n \) edges yields a cyclic \( G \)-decomposition of \( K_{2nx+1} \) for every positive integer \( x \). We extend the concept of an ordered \(\rho\)-labeling to bipartite digraphs and show that an ordered directed \(\rho\)-labeling of a bipartite digraph \( D \) with \( n \) arcs yields a cyclic \( D \)-decomposition of \( K_{nx+1}^* \) for every positive integer \( x \). We also find several classes of bipartite digraphs that admit an ordered directed \(\rho\)-labeling.