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.

Compressed sensing (CS), which is a rising technique of signal processing, successfully manages the huge expenditure of increasing the sampling rate as well as the intricate issues to our work. Hence, more and more attention has been paid to CS during recent years. In this paper, we construct a family of error-correcting pooling designs based on singular linear space over finite fields, which can be efficiently applied to signal processing in terms of CS.

It is proven that for all positive integers \( k \), \( n \), and \( r \), every sufficiently large positive integer is the sum of \( r \) or more \( k \)th powers of distinct elements of \(\{n,n + 1,n + 2,\ldots\}\). The case \( n = 1 \) is the conjecture in the title of [1].

In 1770, Waring conjectured that for each positive integer \( k \) there exists a \( g(k) \) such that every positive integer is a sum of \( g(k) \) or fewer \( k \)th powers of positive integers. Hilbert proved this theorem in 1909, giving rise to Waring’s problem, which asks, for each \( k \), what is the smallest \( g(k) \) such that the statement holds. For further details, see [3].

As a natural question arising from this problem, Johnson and Laughlin [1] proposed what they called an anti-Waring conjecture, which is the following: If \( k \) and \( r \) are positive integers, then every sufficiently large positive integer is the sum of \( r \) or more distinct \( k \)th powers of positive integers. When this holds for a pair \( k, r \), let \( N(k,r) \) denote the smallest positive integer such that each integer \( n \) greater than or equal to \( N(k,r) \) is the sum of \( r \) or more \( k \)th powers of distinct positive integers. As noted in [1], it is easy to see that, for all \( r \), \( N(1,r) = 1 + 2 + \cdots + r = \frac{r(r+1)}{2} \). It is also shown in [1] that \( N(2,1) = N(2, 2) = N(2,3) = 129 \).

Johnson and Laughlin further posed the question of whether given any positive integers \( k \), \( n \), \( r \), there exists an integer \( N(k,n,r) \) such that every integer \( z \) greater than or equal to \( N(k,n,r) \) can be written as a sum of \( r \) or more distinct elements from the set \( \{m^k \mid m \in \mathbb{N}, m \geq n\} \). The aim of this paper is to prove both this statement and the anti-Waring conjecture to be true.

We consider an optimization problem motivated by the tradeoff between connectivity and resilience in key predistribution schemes (KPS) for sensor networks that are based on certain types of combinatorial designs. For a specific class of designs, we show that there is no real disadvantage in requiring the underlying design to be regular.