The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...

A graph is said to be determined by the adjacency spectrum (DS for short) if there is no other nonisomorphic graph with the same spectrum. All connected graphs with index at most √ 2 + √ 5 are known. In this paper, we show that with few exceptions all of these graphs are DS. AMS Subject Classification: 05C50.

Let be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ − cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with . Using the minimal polynomial of A and the theory of Smith normal forms we first determine which p-ranks of A follow directly from the ...

We present a graph spectral approach in which melodies are represented as graphs, based on the intervals between the notes they are composed of. These graphs are then indexed into a database using their laplacian spectra as a feature vector. This laplacian spectrum is a good representative of the original melody. Consequently, range searching around the query spectrum returns similar melodies.

the first zagreb index $m_1$ of a graph $g$ is equal to the sum of squaresof degrees of the vertices of $g$. goubko proved that for trees with $n_1$pendent vertices, $m_1 geq 9,n_1-16$. we show how this result can beextended to hold for any connected graph with cyclomatic number $gamma geq 0$.in addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomaticnumber $gamma$, and minimal $m...

the total version of geometric–arithmetic index of graphs is introduced based on the endvertexdegrees of edges of their total graphs. in this paper, beside of computing the total gaindex for some graphs, its some properties especially lower and upper bounds are obtained.

objectives: this study compared the effects of individual and group training on general health and stress of parents of children with autism spectrum disorders. methods: a sample of 75 parents of children referring to autistic children foundation (including 50 parents in two experimental groups and 25 parents in control group) was taken into account. sample parents were selected through accessi...

a graph that contains a hamiltonian cycle is called a hamiltonian graph. in this paper wecompute the first and the second geometric – arithmetic indices of hamiltonian graphs. thenwe apply our results to obtain some bounds for fullerene.