We develop an ordering function on the class of tournament digraphs (complete antisymmetric digraphs) that is based on the Rényi \(\alpha\)-entropy. This ordering function partitions tournaments on \(n\) vertices into equivalence classes that are approximately sorted from transitive (the arc relation is transitive — the score sequence is \((0, 1, 2, \ldots, n-1)\)) to regular (score sequence like \((\frac{n-1}{2}, \ldots, \frac{n-1}{2})\)). However, the diversity among regular tournaments is significant; for example, there are 1,123 regular tournaments on 11 vertices and 1,495,297 regular tournaments on 13 vertices up to isomorphism, which is captured to an extent.

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem. In this paper, we develop an algorithm for computing the pansophy of graphs and illustrate the algorithm on graphs where the pansophy is already known. We close with future research directions.

Let \( G \) be one of the two multigraphs obtained from \( K_4-e \) by replacing two edges with a double-edge while maintaining a minimum degree of 2. We find necessary and sufficient conditions on \( n \) and \(\lambda\) for the existence of a \( G \)-decomposition of \( K_n \).

Let \( m \) and \( n \) be positive integers, and let \( R = (r_1, \ldots, r_m) \) and \( S = (s_1, \ldots, s_n) \) be non-negative integral vectors. Let \( \mathcal{A}(R, S) \) be the set of all \( m \times n \) \((0,1)\)-matrices with row sum vector \( R \) and column vector \( S \). Let \( R \) and \( S \) be non-increasing, and let \( F(R, S) \) be the \( m \times n \) \((0,1)\)-matrix where for each \( i \), the \( i^{\text{th}} \) row of \( F(R, S) \) consists of \( r_i \) 1’s followed by \( n – r_i \) 0’s, called Ferrers matrices. The discrepancy of an \( m \times n \) \((0,1)\)-matrix \( A \), \( \text{disc}(A) \), is the number of positions in which \( F(R, S) \) has a 1 and \( A \) has a 0. In this paper we investigate linear operators mapping \( m \times n \) matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular we characterize linear operators that preserve both the set of Ferrers matrices and the set of matrices of discrepancy one.

The line graph \( L(G) \) of a nonempty graph \( G \) has the set of edges in \( G \) as its vertex set where two vertices of \( L(G) \) are adjacent if the corresponding edges of \( G \) are adjacent. Let \( k > 2 \) be an integer and let \( G \) be a graph containing k-paths (paths of order \( k \)). The k-path graph \( P_k(G) \) of \( G \) has the set of k-paths of \( G \) as its vertex set where two distinct vertices of \( P_k(G) \) are adjacent if the corresponding k-paths of \( G \) have a \( (k-1) \)-path in common. Thus, \( P_2(G) = L(G) \) and \( P_3(G) = L(L(G)) \). Hence, the k-path graph \( P_k(G) \) of a graph \( G \) is a generalization of the line graph \( L(G) \). Let \( G \) be a connected graph of order \( n > 3 \) and let \( k \) be an integer with \( 2 < k 2 \), then \( P_3(T) \) is \( k \)-tree-connected. This conjecture was verified for \( k = 2, 3 \). In this work, we show that if \( T \) is a tree of order at least 6 containing no vertices of degree 2, 3, 4, or 5, then \( P_3(T) \) is 4-tree-connected and so verify the conjecture for the case when \( k = 4 \).

We define the \( (i, j) \)-liars’ domination number of \( G \), denoted by \( LR(i, j)(G) \), to be the minimum cardinality of a set \( L \subseteq V(G) \) such that detection devices placed at the vertices in \( L \) can precisely determine the set of intruder locations when there are between 1 and \( i \) intruders and at most \( j \) detection devices that might “lie”.

We also define the \( X(c_1, c_2, \ldots, c_t, \ldots) \)-domination number, denoted by \( \gamma _{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \), to be the minimum cardinality of a set \( D \subseteq V(G) \) such that, if \( S \subseteq V(G) \) with \( |S| = k \), then \( |(\bigcup_{v \in S} N[v]) \cap D| \geq c_k \). Thus, \( D \) dominates each set of \( k \) vertices at least \( c_k \) times making \( \gamma_{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \) a set-sized dominating parameter. We consider the relations between these set-sized dominating parameters and the liars’ dominating parameters.

For Cauchy numbers of the first kind \( \{a_n\}{n \geq 0} \) and Cauchy numbers of the second kind \( \{b_n\}{n \geq 0} \), this paper focuses on the log-convexity of some sequences related to \( \{a_n\}{n \geq 0} \) and \( \{b_n\}{n \geq 0} \). For example, we discuss log-convexity of \( \{n|a_n| – |a_{n+1}|\}{n \geq 1} \), \( \{b{n+1} – nb_n\}{n \geq 1} \), \( \{n|a_n|\}{n \geq 1} \), and \( \{(n + 1)b_n\}_{n \geq 0} \). In addition, we investigate log-balancedness of some sequences involving \( a_n \) (or \( b_n \)).

Let \( G \) be a graph. We define the distance \( d \) pebbling number of \( G \) to be the smallest number \( s \) such that if \( s \) pebbles are placed on the vertices of \( G \), then there must exist a sequence of pebbling moves which takes a pebble to a vertex which is at a distance of at least \( d \) from its starting point. In this article, we evaluate the distance \( d \) pebbling numbers for a directed cycle graph with \( n \) vertices.

Let \( G \) be a \( k \)-connected (\( k \geq 2 \)) graph of order \( n \). If \( \gamma(G^c) \geq n – k \), then \( G \) is Hamiltonian or \( K_k \vee K_{k+1}^c \), where \( \gamma(G^c) \) is the domination number of the complement of the graph \( G \).

An \emph{Italian dominating function} on a digraph \( D \) with vertex set \( V(D) \) is defined as a function \( f : V(D) \to \{0, 1, 2\} \) such that every vertex \( v \in V(D) \) with \( f(v) = 0 \) has at least two in-neighbors assigned 1 under \( f \) or one in-neighbor \( w \) with \( f(w) = 2 \). The weight of an Italian dominating function is the sum \( \sum_{v \in V(D)} f(v) \), and the minimum weight of an Italian dominating function \( f \) is the \emph{Italian domination number}, denoted by \( \gamma_I(D) \). We initiate the study of the Italian domination number for digraphs, and we present different sharp bounds on \( \gamma_I(D) \). In addition, we determine the Italian domination number of some classes of digraphs. As applications of the bounds and properties on the Italian domination number in digraphs, we give some new and some known results of the Italian domination number in graphs.