A design graph is a regular bipartite in which any two distinct vertices of the same part have number common neighbors. This class graphs has close relationship to strongly graphs. In this paper, we study distance eigenvalues Also, will explicitly determine graphs, and values for integral, that is, its are integers.

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

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For two graphs $mathrm{G}$ and $mathrm{H}$ with $n$ and $m$ vertices, the corona $mathrm{G}circmathrm{H}$ of $mathrm{G}$ and $mathrm{H}$ is the graph obtained by taking one copy of $mathrm{G}$ and $n$ copies of $mathrm{H}$ and then joining the $i^{th}$ vertex of $mathrm{G}$ to every vertex in the $i^{th}$ copy of $mathrm{H}$. The neighborhood corona $mathrm{G}starmathrm{H}$ of $mathrm{G}$ and $...

We characterize the rational subsets of a finite group and discuss the relations to integral Cayley graphs.

In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a perfect state transfer (PST) between vertices a, b ∈ V (G) if |F (τ)ab| = 1, for some positive real number τ ...