Article history: Received 14 April 2013 Available online 5 March 2014

In this paper, we completely describe the spectrum of the generalized Petersen graph P (n, k), thus adding to the classes of graphs whose spectrum is completely known.

In 1995, Brouwer proved that the toughness of a connected k-regular graph G is at least k/λ − 2, where λ is the maximum absolute value of the non-trivial eigenvalues of G. Brouwer conjectured that one can improve this lower bound to k/λ − 1 and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to k/λ. In this pape...

The spectral theory of quantum graphs is related via an exact trace formula with the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e., finding out how many different lengths exist for periodic orbits with a given period and the average number of periodic orbits with the same length) is necessary for the...

The tenacity of an incomplete connected graph G is defined as T (G) = min{ |S|+m(G−S) ω(G−S) : S ⊂ V (G), ω(G− S) > 1}, where ω(G− S) and m(G− S), respectively, denote the number of components and the order of a largest component inG−S. This is a reasonable parameter to measure the vulnerability of networks, as it takes into account both the amount of work done to damage the network and how bad...

Topoi are known to be categories with extra properties that make them much alike the category of Sets. In a Topoi it is possible to define adequate notions of membership, elements and subobjects, power ”sets”, and finally, every Topoi has an internal logic able to justify any reasoning carried inside it. Most of the cases, this logic is not Classical (Boolean). The general logic for the Topoi i...

Article history: Available online 25 March 2011 Submitted by J.J. Loiseau AMS classification: 15A80 15A39 15A03

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class...

In this paper we introduce the k’th isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of k disjoint subsets of the vertex set of the graph. In this direction we show that the second isoperimetric constant in the general setting coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of th...

When can one see from the spectrum of a graph whether it is distance-regular or not? We give some new results for when this is the case. As a consequence we find (among others) that the following distance-regular graphs are uniquely determined by their spectrum: The collinearity graphs of the generalized octagons of order (2, 1), (3, 1) and (4, 1), the Biggs-Smith graph, the M22 graph, and the ...