This paper presents tight bounds and characterizations for the vertex cover number and the connected vertex cover number of graphs. In particular, we identify all graphs for which βc(G) = |V (G)| − 1, proving that these are exactly the cycles and complete graphs. The analysis employs tools such as the degree matrix and the Rayleigh quotient to derive new and sharp upper bounds.
For a planar graph G of order n, let F(G) be the set of all faces of G embedded into R 2 , including the exterior face. A bijective vertex labeling f : V (G) → {1, 2, ..., n} induces a face labeling f ∗ : F(G) → N defined by setting f ∗ (F) equal to the sum of all labels of the boundary vertices of F. The graph G is said to be hyper face-magic if there exists a vertex labeling whose induced face labeling is constant. In this paper, we state properties of hyper face-magic graphs and construct various classes of hyper face-magic graphs.
Let \(G=(V,E)\) be a connected, finite undirected graph. A set \(S \subseteq V\) is said to be a total dominating set of \(G\) if every vertex in \(V\) is adjacent to some vertex in \(S\). The total domination number, \(\gamma_{t}(G)\), is the minimum cardinality of a total dominating set in \(G\). We define the \(k\)-total bondage of $G$ to be the minimum number of edges to remove from \(G\) so that the resulting graph has a total domination number at least \(k\) more than \(\gamma_{t}(G)\). In this work we establish general properties of \(k\)-total bondage and find exact values for certain graph classes including paths, cycles, wheels, complete and complete bipartite graphs.
This project mathematically models the self-assembly of DNA nanostructures in the shape of select Archimedean graphs using the flexible tile model. Under three different sets of restrictions called scenarios, we employ principles of linear algebra and graph theory to determine the minimum number of different DNA branched molecules and bond types needed to construct the desired shapes, theoretically reducing laboratory costs and the waste of biomaterials. We determine exact values for T_3(G), the minimum number of molecule (or “tile") types needed for all six order 12 and 24 Archimedean graphs. We also determine exact values for B_3(G), the minimum number of strand (or “bond-edge") types, for three of the six graphs and establish bounds for the remaining three. Two algorithms, implemented as Python scripts, are used to analyze proposed design strategies for the graphs.
The complexity of a graph is the number of its labeled spanning trees. It is demonstrated that the seven known triangle-free strongly regular graphs, such as the Higman-Sims graph, are graphs of maximal complexity among all graphs of the same order and degree; their complements are shown to be of minimal complexity. A generalization to nearly regular graphs with two distinct eigevalues of the Laplacian is presented. Conjectures and applications of these results to biological problems on neuronal activity are described.
