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In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular for lollipop graphs and generalized barbell graphs. For a matrix A with associated graph G , a new technique utilizing the strong spectral property is introduced, allowing us to construct a matrix A ′ whose graph is obtained from G by appending a clique while arbitrary list of eigenvalues is added to the spectrum. Consequently, many spectra are shown realizable for block graphs. [ABSTRACT FROM AUTHOR]

We introduce the set G SSP of all simple graphs G with the property that each symmetric matrix corresponding to a graph G ∈ G SSP has the strong spectral property. We find several families of graphs in G SSP and, in particular, characterise the trees in G SSP. [ABSTRACT FROM AUTHOR]

In this paper we introduce a parameter Mm ( G ) , defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph G . We develop basic properties of Mm ( G ) and compute it for several families of graphs. [ABSTRACT FROM AUTHOR]

Abstract: Let G be an undirected graph on n vertices and let be the set of all real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The Inverse Eigenvalue Problem for a graph G is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in . This question is, in general, hard to answer and several variations were studied, most notably the minimum rank problem. In this paper we introduce the problem of determining for which graphs G there exists a matrix in whose characteristic polynomial is a square, i.e. the multiplicities of all its eigenvalues are even. We solve this question for several families of graphs. [Copyright &y& Elsevier]

In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle. [ABSTRACT FROM AUTHOR]