Mathematics & Statistics Faculty

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Browsing Mathematics & Statistics Faculty by Author "Alinaghipour, Fatemeh"

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    Minimum number of distinct eigenvalues of graphs
    (International Linear Algebra Society, 2013-09) Ahmadi, Bahman; Alinaghipour, Fatemeh; Cavers, Michael; Fallat, Shaun; Meagher, Karen; Nasserasr, Shahla
    The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.
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    On the relationship between zero forcing number and certain graph coverings
    (2014) Alinaghipour, Fatemeh; Fallat, Shaun; Meagher, Karen
    The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block- cycle graph the zero forcing number equals the path cover number. We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.