Eigenvalues of a H-generalized join graph operation constrained by vertex subsets

Considering a graph $H$ of order $p$, a generalized $H$-join operation of a family of graphs $G_1,..., G_p$, constrained by a family of vertex subsets $S_i \subseteq V(G_i)$, $i=1,..., p,$ is introduced. When each vertex subset $S_i$ is $(k_i,τ_i)$-regular, it is deduced that all non-main adjacency eigenvalues of $G_i$, different from $k_i-τ_i$, for $i=1,..., p,$ remain as eigenvalues of the graph $G$ obtained by the above mentioned operation. Furthermore, if each graph $G_i$ of the family is $k_i$-regular, for $i=1,..., p$, and all the vertex subsets are such that $S_i=V(G_i)$, the $H$-generalized join operation constrained by these vertex subsets coincides with the $H$-generalized join operation. Some applications on the spread of graphs are presented. Namely, new lower and upper bounds are deduced and a infinity family of non regular graphs of order $n$ with spread equals $n$ is introduced.

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