A new method for constructing graphs determined by their generalized spectrum

Abstract

A graph G is said to be determined by its generalized spectrum (DGS for short) if for any graph H, H and G are cospectral with cospectral complements implies that H is isomorphic to G. Let Ĝ be the graph obtained from G by adding a pendent edge at every vertex of G. We show that Ĝ is DGS if and only if G is DGS for some graph G. This gives a simple way to construct large DGS graphs from small ones explicitly. In particular, we show that every graph in the infinite sequence G, Ĝ , Ĝ , · · · is DGS, for some DGS graph G.

Construction of Sc Chordal and Sc Weakly Chordal Graphs

“…The overall procedure of the construction of sc chordal graph with the given degree sequence (10,10,8,8,6, 6, 5, 5, 3, 3, 1, 1) is obtained as given below and illustrated in figure-1 (i). The graph G 2 as shown in figure-1 respectively.…”

“…Since vertex v 6 already got 5 adjacency, which can be seen in graph G 6 . So we have to give only 1 adjacency in the form of adding edge e 33 from vertex v 6 to v 8 . In this way we obtain graph G 7 as shown in figure-1(g).…”

“…Figure 1. Construction of sc chordal graph with the given degree sequence (10,10,8,8,6,6,5,5,3,3,1,1) …”

“…There are several techniques discussed in [2], [8] and [13] for the construction of a graphs having some specific properties. In [13] X. Huang and Q. Huang gave a method to construct many classes of connected graphs with exactly kmain eigenvalues for any positive integer k. In [8] Lihuan Mao et.…”

“…In [13] X. Huang and Q. Huang gave a method to construct many classes of connected graphs with exactly kmain eigenvalues for any positive integer k. In [8] Lihuan Mao et. al gave a new method for the construction of graphs by using their generalized spectrum.…”