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.