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The classical problem of characterizing the graphs whose eigenvalues lie in a given interval may date back to the work of J.H. Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than −2 has attracted widespread attention. Mixed graphs are natural generalizations of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than − 5 + 1 2. In fact, we found three infinite classes of mixed graphs and 30 scattered mixed graphs. [ABSTRACT FROM AUTHOR]

Recently, Gnutzmann and Smilansky [5] presented a formula for the bond scattering matrix of a graph with respect to an Hermitian matrix. We present another proof for this formula by a technique use in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an L -function of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its L -functions. [ABSTRACT FROM AUTHOR]

Abstract: When an edge is removed from an undirected graph, there is a limited change that can occur in the multiplicity of an eigenvalue of a Hermitian matrix with that graph. Primarily for trees, we identify the changes that can occur and characterize the circumstances under which they occur. This extends known results for the removal of vertices. A catalog of examples is given to illustrate the possibilities that can occur and to contrast the case of trees with that of general graphs. [ABSTRACT FROM AUTHOR]

Let d , n ∈ Z + such that 1 ≤ d ≤ n. A d -code C ⊂ F q n × n is a subset of order n square matrices with the property that for all pairs of distinct elements in C , the rank of their difference is greater than or equal to d. A d -code with as many as possible elements is called a maximum d -code. The integer d is also called the minimum distance of the code. When d < n , a classical example of such an object is the so-called generalized Gabidulin code (Gabidulin and Kshevetskiy (2005)). In Delsarte and Goethals (1975), Schmidt (2015) and K.U. Schmidt (2018), several classes of maximum d -codes made up respectively of symmetric, alternating and Hermitian matrices were exhibited. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric 2-code which is not equivalent to the one with same parameters constructed in Schmidt (2015). [ABSTRACT FROM AUTHOR]