Narrow Results

A social group is represented by a graph, where each pair of nodes is connected by two oppositely directed links. At the beginning, a given amount p(i) of resources is assigned randomly to each node i. Also, each link r(i,j) is initially represented by a random positive value, which means the percentage of resources of node i which is offered to node j. Initially, the graph is fully connected, i.e., all nondiagonal matrix elements r(i,j) are different from zero. During the simulation, the amounts of resources p(i) change according to the balance equation. The nodes reorganize their activity with time to give more resources to those which give them more. This is the rule of varying the coefficients r(i,j). The result is that after some transient time, only some pairs (m,n) of nodes survive with nonzero p(m) and p(n), each pair with symmetric and positive r(m,n)=r(n,m). Other coefficients r(m,i≠n) vanish. Unpaired nodes remain with no resources, i.e., their p(i)=0, and they cease to be active as they have nothing to offer. The percentage of survivors (i.e., those with p(i) positive) increases with the velocity of varying the numbers r(i,j), and it slightly decreases with the size of the group. The picture and the results can be interpreted as a description of a social algorithm leading to marriages.

In this paper we introduce a new class of cooperative games. We define a characteristic function over the cycles of a digraph. We present a mathematical model for this situation and an axiomatic characterization of a solution for this class of cooperative games. This is introduced as a method to measure the importance of the nodes in a digraph, and can be related with the Shapley value of a game in characteristic function form. Also, we extend the modeling by applying coalitional structures for the nodes and r-efficient solutions, where the allocation amount is a real number r, showing axiomatic solutions in both cases.

In this work, we investigate the structure of the digraph associated with the square mapping on the ring of Gaussian integers by using the exponent of the unit group modulo γ. The formula for the fixed points of is established. Some connections of the lengths of cycles with the exponent of the unit group modulo γ are presented. Furthermore, we study the maximum distance from the cycle on each component.

Let m_t(α) denote the t-metric Mahler measure of the algebraic number α. Recent work of the first author established that the infimum in m_t(α) is attained by a single point for all sufficiently large t. Nevertheless, no efficient method for locating is known. In this paper, we define a new tree data structure, called a factorization tree, which enables us to find when α ∈ ℚ. We establish several basic properties of factorization trees, and use these properties to locate in previously unknown cases.

Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eigenvalues. Extremal energy and iota energy of unicyclic signed digraphs is known. In this paper, we address the problem of finding signed digraphs with extremal iota energy among vertex-disjoint bicyclic signed digraphs of fixed order.

The energy of a weighted digraph $D$ is the sum of absolute values of real part of its eigenvalues. Recently, the minimal and maximal energy of unicyclic weighted digraphs with cycle weight $r\in [-1,1]\setminus \{0\}$ is studied. In this paper, we introduce a class $𝔹(r,s)$ of those bicyclic weighted digraphs of fixed order which contain vertex-disjoint weighted cycles of weights $r$ or $-r$ and $s$ or $-s$, where $0<r,s\le 1$. We find digraphs in $𝔹(r,s)$ under certain conditions on $r$ and $s$ with extremal energy.

The iota energy of an $n$-vertex digraph $D$ is defined by $E_c(D)={\sum }_{k=1}^n|\mathrm{\text{Im}}(z_k)|$, where $z_1,\dots ,z_n$ are eigenvalues of $D$ and $\mathrm{\text{Im}}(z_k)$ is the imaginary part of eigenvalue $z_k.$ Recently, this concept of iota energy of digraphs is extended to sidigraphs. The iota energy of a sidigraph is defined in an analogous way. In this paper, we consider a class of digraphs with two linear subdigraphs of equal length and find digraphs in this class with extremal iota energy. Furthermore, we take a similar class of sidigraphs and find minimal and maximal iota energy among the sidigraphs in this class.

We introduce a new notion of convexity in digraphs, which we call incoming-path convexity, and prove that the incoming-path convexity space of a digraph is a convex geometry (that is, it satisfies the Minkowski–Krein–Milman property) if and only if the digraph is acyclic. Moreover, we prove that incoming-path convexity is adequate to characterize collapsibility of models generated by Bayesian networks. Based on these results, we also provide simple linear algorithms to solve two topical problems on Markov properties of a Bayesian network (that is, on conditional independences valid in a Bayesian network).

Let $D$ be a digraph of order $n$ and let $A(D)$ be the adjacency matrix of $D.$ Let Deg$(D)$ be the diagonal matrix of vertex out-degrees of $D.$ For any real $\alpha \in [0,1]$ the generalized adjacency matrix $A_{\alpha }(D)$ of the digraph $D$ is defined as $A_{\alpha }(D)=\alpha \mathrm{\text{Deg}}(D)+(1-\alpha )A(D).$ This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of $D$. In this paper, we find $A_{\alpha }$-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the $A_{\alpha }$-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the $A_{\alpha }$-spectrum of various families of unsymmetric digraphs.

Let $D$ be a directed graph with $n$ vertices and $m$ arcs. A function $g:V(D)\to \{1,2,\dots ,k\}$ where $k\le n$ is called a complete coloring of $D$ if and only if for every arc $uv$ of $D$, the ordered pair $(g(u),g(v))$ appears at least once. If the pair $(i,i)$ is not assigned, then $g$ is called a $proper$ $complete$ $coloring$ of $D$. The maximum $k$ for which $D$ admits a proper complete coloring is called the $achromatic$ $number$ of $D$ and is denoted by $\overrightarrow{{\psi }_c}(D)$. We obtain the upper bound for the achromatic number of digraphs and regular digraphs and investigate the same for some classes of digraphs such as unipath, unicycle, circulant digraphs, etc.

Let $D$ be a digraph of order $n$. Let $A(D)$ and $\mathrm{\text{Deg}}(D)$ be respectively, the adjacency matrix and the diagonal matrix of vertex out-degrees of $D.$ The generalized adjacency matrix $A_{\alpha }(D)$ of $D$ is defined as