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In this paper, we study the inverse eigenvalue problem of constructing symmetric matrices whose graph is a tree, i.e. of constructing irreducible acyclic matrices from given eigendata consisting of the smallest and largest eigenvalues of their leading principal submatrices. We solve the problem completely for the case when the given eigenvalues are distinct and the subgraph of the tree induced by { 1 , 2 , ... , j } remains connected for 1 ≤ j ≤ n. The result generalizes the results known for matrices described by some specific trees. The proofs of the main results are constructive and so provide an algorithm for computing the actual entries of the required matrix. Numerical solutions of the problem are then obtained with SCILAB by feeding the eigendata and plugging in the adjacency matrix as inputs. We also make an attempt to find the number of solutions for several special families of unlabelled trees. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as L E = ∑ i = 1 n | μ i - 2 m n | LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|} , L E L = ∑ i = 1 n - 1 μ i LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} } and K f = n ∑ i = 1 n - 1 1 μ i Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf. [ABSTRACT FROM AUTHOR]

In 2022, the second author found a prolific construction of strongly regular graphs, which is based on joining a coclique and a divisible design graph with certain parameters. The construction produces strongly regular graphs with the same parameters as the complement of the symplectic graph Sp (2 d , q) . In this paper, we determine the parameters of strongly regular graphs which admit a decomposition into a divisible design graph and a coclique attaining the Hoffman bound. In particular, it is shown that when the least eigenvalue of such a strongly regular graph is a prime power, its parameters coincide with those of the complement of Sp (2 d , q) . Furthermore, a generalization of the construction is discussed. [ABSTRACT FROM AUTHOR]

In this paper we first provide two new constructions for Cayley sum graphs, namely, norm-coset graphs and trace-coset graphs, and determine their second largest eigenvalues using Gaussian sums. Next, a connection between Cayley sum graphs and complex codebooks is established. Based on this, infinite families of asymptotically optimal complex codebooks are explicitly constructed. The derived Cayley sum graphs and codebooks either include some known constructions as special cases or provide flexible new parameters. [ABSTRACT FROM AUTHOR]

A conference matrix of order n is an n × n matrix C with diagonal entries 0 and off-diagonal entries ± 1 satisfying C C ⊤ = (n - 1) I . If C is symmetric, then C has a symmetric spectrum Σ (that is, Σ = - Σ ) and eigenvalues ± n - 1 . We show that many principal submatrices of C also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction. [ABSTRACT FROM AUTHOR]