ALAMT  Vol.4 No.2 , June 2014
On the Spectral Characterization of H-Shape Trees
Abstract: A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. An H-shape is a tree with exactly two of its vertices having maximal degree 3. In this paper, a formula of counting the number of closed 6-walks is given on a graph, and some necessary conditions of a graph Γ cospectral to an H-shape are given.
Cite this paper: Hu, S. (2014) On the Spectral Characterization of H-Shape Trees. Advances in Linear Algebra & Matrix Theory, 4, 79-86. doi: 10.4236/alamt.2014.42005.

[1]   Cvetkovi'c, D., Doob, M. and Sachs, H. (1980) Spectra of Graphs—Theory and Application. Academic Press, New York.

[2]   van Dam, E.R. and Haemers, W.H. (2003) Which Graph Are Determined by Their Spectrum? Linear Algebra and Its Applications, 373, 241-272.

[3]   Doob, M. and Haemers, W.H. (2002) The Complement of the Path Is Determined by Its Spectrum. Linear Algebra and Its Applications, 356, 57-65.

[4]   Noy, M. (2003) Graphs Determined by Polynomial Invariants. Theoretical Computer Science, 307, 365-384.

[5]   Smith, J.H. (1970) Some Propertice of the Spectrum of Graph. In: Guy, R., et al., Eds., Combinatorial Structure and Their Applications, Gordon and Breach, New York, 403-406.

[6]   Wang, W. and Xu, C.-X. (2006) On the Spactral Characterization of T-Shape Trees. Linear Algebra and Its Applications, 414, 492-501.

[7]   Schwenk, A.J. (1973) Almost All Trees Are Cospectral. In: Harary, F., Ed., New Directions in the Theory of Graphs, Academic Press, New York, 275-307.

[8]   Godsil, C.D. (1993) Algebraic Combinatorics. Chapman & Hall, New York.

[9]   Sachs, H. (1964) Beziehungen zwischen den in einem graphen enthaltenen kreisenund seinem charakteristischen polynom. Publicationes Mathematicae, 11, 119-134.

[10]   Omidi, G.R. and Tajbakhsh, K. (2007) Starlike Trees Are Determined by Their Laplacian Spectrum. Linear Algebra and Its Applications, 422, 654-658.