On the Colin de Verdière number of graphs

Colin de Verdiere parameters of chordal graphs

“…If G is a split graph, then µ(G) = tw(G) + 1 if and only if G is Type II. They also proved We now determine µ(G) for all simple chordal graphs, thus extending the result in [8] for split graphs, by first establishing a relationship between µ(G) and µ(G⊖ µ v). Proof.…”

“…Utilizing an operation called orthogonal removal, which generalizes to vertices of any degree the ∆Y and Y∆ transforms (see [17]), we determine precisely when a chordal simple graph can take on each of the values allowed for both µ and ν. For the case of µ, this extends and makes use of precise results for split graphs from the recent work in [8]. A byproduct of our analysis is a precise result for ν of chordal multigraphs.…”

“…The parameters µ and ν have been studied extensively over the past twenty years and have arisen in the context of vertex connectivity [19,20] and in connection with cliques and chordal graphs [8,16]. Colin de Verdière's original definition of ν was for simple graphs [7], and this definition was extended to a multigraph setting (and to complex matrices) in [12,15].…”

“…For chordal graphs it is known (see [8,12,13,15]) that both µ and ν can take on only one of two values (depending on the tree-width or maximum clique size). Utilizing an operation called orthogonal removal, which generalizes to vertices of any degree the ∆Y and Y∆ transforms (see [17]), we determine precisely when a chordal simple graph can take on each of the values allowed for both µ and ν.…”

“…Split graphs are exactly those simple chordal graphs whose complement is chordal. In [8], µ was determined for the case of split graphs: A split graph with maximum clique C and independent set S is Type II if there exist vertices v ∈ C and u 1 , u 2 ∈ S with N (u 1 ) = N (u 2 ) = C − v, otherwise it is called Type I. If G is a split graph, then µ(G) = tw(G) + 1 if and only if G is Type II.…”