Integral Mixed Cayley Graphs over Abelian Groups

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

C-graphs - A Mixed Graphical Representation of Groups

Corresponding to each group Γ, a mixed graph G = (Γ,E,E′) called C-graph is assigned, such that the vertex set of G is the group itself. Two types of adjacency relations, that is, one way and two way communication is defined for vertices, to get a clear idea of the underlying group structure. An effort to answer the question, ‘Is there any relation between the order of an element in the group and degrees of the corresponding vertex in the C-graph’, by proposing a mathematical formula connecting them is made. Established an upper bound for the total number of edges in a C-graph G. For a vertex z in G, the concept Connector Edge CEz is defined, which convey some structural properties of the group Γ. The Connector Edge Set is defined for both a vertex z and the whole C-graph G, and is denoted as C E z and C E G respectively. Proposed the result, C E G = E if and only if |Γ| = 2n, n ∈ N. Finally, the properties of G, which the Connector Edge Set C E G carry out are discussed.

Scheduling Multiprocessor Tasks with Equal Processing Times as a Mixed Graph Coloring Problem

This article extends the scheduling problem with dedicated processors, unit-time tasks, and minimizing maximal lateness for integer due dates to the scheduling problem, where along with precedence constraints given on the set of the multiprocessor tasks, a subset of tasks must be processed simultaneously. Contrary to a classical shop-scheduling problem, several processors must fulfill a multiprocessor task. Furthermore, two types of the precedence constraints may be given on the task set . We prove that the extended scheduling problem with integer release times of the jobs to minimize schedule length may be solved as an optimal mixed graph coloring problem that consists of the assignment of a minimal number of colors (positive integers) to the vertices of the mixed graph such that, if two vertices and are joined by the edge , their colors have to be different. Further, if two vertices and are joined by the arc , the color of vertex has to be no greater than the color of vertex . We prove two theorems, which imply that most analytical results proved so far for optimal colorings of the mixed graphs , have analogous results, which are valid for the extended scheduling problems to minimize the schedule length or maximal lateness, and vice versa.

Mixed graph colouring as scheduling multi-processor tasks with equal processing times

A problem of scheduling partially ordered unit-time tasks processed on dedicated machines is formulated as a mixed graph colouring problem, i. e., as an assignment of integers (colours) {1, 2, …, t} to the vertices (tasks) V {ν1, ν2, …, νn}, of the mixed graph G = (V, A, E) such that if vertices vp and vq are joined by an edge [νp, νq] ∈ E their colours have to be different. Further, if two vertices νp and νq are joined by an arc (νi, νj) ∈ A the colour of vertex νi has to be no greater than the colour of vertex νj. We prove that an optimal colouring of a mixed graph G = (V, A, E) is equivalent to the scheduling problem GcMPT|pi = 1|Cmax of finding an optimal schedule for partially ordered multi-processor tasks with unit (equal) processing times. Contrary to classical shop-scheduling problems, several dedicated machines are required to process an individual task in the scheduling problem GcMPT|pi = 1|Cmax. Moreover, along with precedence constraints given on the set V {ν1, ν2, …, νn}, it is required that a subset of tasks must be processed simultaneously. Due to the theorems proved in this article, most analytical results that have been proved for the scheduling problems GcMPT |pi = 1|Cmax so far, have analogous results for optimal colourings of the mixed graphs G = (V, A, E), and vice versa.

The Hermitian Kirchhoff Index and Robustness of Mixed Graph

Large-scale social graph data poses significant challenges for social analytic tools to monitor and analyze social networks. The information-theoretic distance measure, namely, resistance distance, is a vital parameter for ranking influential nodes or community detection. The superiority of resistance distance and Kirchhoff index is that it can reflect the global properties of the graph fairly, and they are widely used in assessment of graph connectivity and robustness. There are various measures of network criticality which have been investigated for underlying networks, while little is known about the corresponding metrics for mixed networks. In this paper, we propose the positive walk algorithm to construct the Hermitian matrix for the mixed graph and then introduce the Hermitian resistance matrix and the Hermitian Kirchhoff index which are based on the eigenvalues and eigenvectors of the Hermitian Laplacian matrix. Meanwhile, we also propose a modified algorithm, the directed traversal algorithm, to select the edges whose removal will maximize the Hermitian Kirchhoff index in the general mixed graph. Finally, we compare the results with the algebraic connectivity to verify the superiority of the proposed strategy.