In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a vertex-clique incidence matrix for a graph and produce new lower bounds for the negative eigenvalues and negative inertia of a graph. Moreover, utilizing these vertex-clique incidence matrices, we generalize several notions such as the signless Laplacian matrix, and develop bounds on the incidence energy and the signless Laplacian energy of the graph. %The tight upper bounds for the energies of a graph and its line graph are given. More generally, we also consider the set $S(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent. An important parameter in this setting is $q(G)$, and is defined to be the minimum number of distinct eigenvalues over all matrices in $S(G)$. For a given graph $G$ the concept of a vertex-clique incidence matrix associated with an edge clique cover is applied to establish several classes of graphs with $q(G)=2$.

The concept of graphs with clique-width at most was first introduced by Courcelle et al.to be the graphs that can be characterized using -expressions derived from graph operations that use labels of vertices. If the clique-width for some graph is bounded then a grand number of algorithmic problems, in general NP-hard, can be solved in polynomial time when restricted to this graph. This important fact motivated the researchers to prove that the clique-width of certain graphs is bounded. Following this research direction, we prove in this paper that the clique-width of series-parallel digraphs is at most 6 and we present an time algorithm to construct a 6-expression for this class of digraphs. In another part, we present a linear time recognition algorithm for a similar class of series-parallel digraphs and prove that the clique-width of this class is at most 3.

We consider the problem of finding the largest clique of a graph. This is an NP-hard problem and no exact algorithm to solve it exactly in polynomial time is known to exist. Several heuristic approaches have been proposed to find approximate solutions, Markov Chain Monte Carlo being one of those. In the context of Markov Chain Monte Carlo, we present a class of ``parallel dynamics'', known as Probabilistic Cellular Automata, which can be used in place of the more standard choice of sequential ``single spin flip'' to sample from a probability distribution concentrated on the largest cliques of the graph. We perform a numerical comparison between the two classes of chains both in terms of the quality of the solution and in terms of computational time. We show that the parallel dynamics are considerably faster than the sequential one while providing solutions of comparable quality.

There has been a great deal of research in the area of using graph engines and graph databases to model network traffic and network attacks, but the novelty of this research lies in visually or graphically representing the Reconnaissance Tactic (TA0043) of the MITRE ATT&CK framework. Using the newly created dataset, UWF-Zeekdata22, based on the MITRE ATT&CK framework, patterns involving network connectivity, connection duration, and data volume were found and loaded into a graph environment. Patterns were also found in the graphed data that match the Reconnaissance as well as other tactics captured by UWF-Zeekdata22. The Star motif was particularly useful in mapping the Reconnaissance tactic. The results of this paper show that graph databases/graph engines can be essential tools for understanding network traffic and trying to detect network intrusions before they happen. Finally, an analysis of the run-time performance of the reduced dataset used to create the graph databases showed that the reduced datasets performed better than the full dataset.