This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Tur\'{a}n graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called \emph{the graphs of pyramids}, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lov\'{a}sz $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.
In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigenvalues are rational and, consequently, that their splitting field is \(\mathbb{Q}\). We then consider supergraphs obtained by adding edges to these graphs and prove that the corresponding splitting field is a quadratic extension of \(\mathbb{Q}\). Next, using their structural decomposition, we examine a general class of Lewis graphs. For this class, we establish bounds on both the number of irrational eigenvalues and the degree of the associated splitting fields. Finally, we investigate prime character degree graphs of diameter \(3\), focusing on the arithmetic nature of their eigenvalues and the degree of their splitting fields.
G. Sivanesan, And C. SELVARAJ, J. Laubacher· 0 citations
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincar\'e-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.
Matthias Hofmann, James B Kennedy, Delio Mugnolo et al.· 0 citations
This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices to contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.
S. Sripriya, A. Anuradha· Baghdad Science Journal· 0 citations
The notion of graph complements has been widely generalized to study diverse structural and spectral properties of graphs. In this paper, we introduce and investigate the concept of generalized color complements of graphs with respect to a prescribed vertex partition. Building on earlier work on generalized color complements, we focus on structural properties arising from the interaction between graph coloring and partition-based complement operations. Sufficient conditions are established under which generalized color complements are disconnected, regular, and Eulerian. Explicit expressions are derived for the degree of any vertex in the generalized color complements [Formula: see text], [Formula: see text]. Furthermore, several classes of self-color-complementary graphs are identified for fixed partitions. A collection of illustrative examples is provided to demonstrate and validate the theoretical results. The findings extend existing results on generalized complements to a color-based framework and contribute to a deeper understanding of partition-dependent graph complements.
S. Sahana, S. D’Souza, S. Nayak et al.· Discrete Mathematics, Algori...· 0 citations
Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.
Vishal Gupta, A. Iosevich, J. Iosevich et al.· 0 citations
A graph category is a category with a set of graphs or similar structures (such as, directed graphs, signed graphs, etc.) playing the role of objects, and an appropriate notion of homomorphism playing the role of morphisms. The characterization of multiplicative objects are important open problems in categories of undirected and directed graphs. While the recent disproving of the Hedetniemi's conjecture due to Shitov (Ann. Math. 2019), which claimed that all complete graphs are multiplicative, provided a breakthrough in the study of multiplicative undirected graphs, the characterization of multiplicative undirected graphs remains known only for cycles, circular cliques $K_{{n/k}}$ where ${n/k} \in (2,4]$, complete graphs, and graphs whose each edge is part of at most one $4$-cycle. Similarly, whether a given directed graph is multiplicative or not is known only for some oriented paths, oriented cycles, and transitive tournaments. We study multiplicative graphs in the category of directed graphs where pushable homomorphism plays the role of morphism. We provide full multiplicativity characterization for directed bipartite graphs, oriented cycles, and transitive tournaments. As a consequence we find new (infinite) classes of non-multiplicative directed graphs in the usual directed graphs category. We also resolve an open question posed by Das \textit{et al.} (CALDAM 2026) related to the existence of exponential directed graphs with respect to pushable homomorphisms, and use our solution as a tool for our proofs.
S. Das, Moritz Muhlenthaler, Sagnik Sen et al.· 0 citations