For a graph G, its k-th graph power G^k is constructed by placing an edge between two vertices if they are within distance k. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral graph theory and linear optimization methods. First, we use the so-called ratio-type bound to provide an alternative and spectral proof of a result by Lovász [IEEE Trans. Inform. Theory 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovász theta number and, hence, also on the Shannon capacity. In fact, we show that Lovász' result holds in the more general context of graph powers. Secondly, we derive another bound on the Shannon capacity of graph powers, the so-called rank-type bound, which depends on a new family of polynomials that can be computed by running a simple algorithm. Lastly, we provide several computational experiments that demonstrate the sharpness of the two proposed algebraic bounds. As a byproduct, when these two new algebraic bounds are tight, they can be used to easily derive the exact values of the Lovász theta number (which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.

We generalize the concept of token graphs to obtain supertoken graphs. In the latter case, there can be more than one token in a vertex. We formally define supertoken graphs and establish their basic properties. Moreover, we provide some bounds and exact values on the independence number, clique number, and chromatic number of these graphs. Finally, we construct a new infinite family of graphs, which we call the p-augmented 2-token graphs of cycles, and study their properties, including the spectral radius or largest adjacency eigenvalue.

The k-token graph F_k(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we describe some properties of the Laplacian matrix L_k of F_k(G) and the Laplacian matrix L_k of the k-token graph F_k(G) of its complement G . First, we provide a complete proof of a result from a previous article on the commutativity of the matrices L_k and L_k . Based on this result, fixed the pair (n, k) and the graph G, we introduce a ‘local’ algebra ℒ(G) , generated by the pair (L_k, L_k) , showing its close relationship with the Bose–Mesner algebra of the Johnson graphs J(n, k). Finally, for fixed (n, k), we present a ‘global’ algebra 𝒜(n,k) that contains ℒ(G) together with the Laplacian and adjacency matrices of the k-token graph of any graph G on n vertices.

The vertices of a k-token graph of a graph G correspond to k indistinguishable tokens placed on k different vertices of G. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of G, we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian k-th power of G by itself) are obtained, and we present some of their properties, including order, size, and connectivity.

In this paper, we study proper and complete edge-colorings of Johnson graphs J(n,2), also called n-triangular graphs. They are isomorphic both to the 2-token graphs of complete graphs and to the line graphs of complete graphs. A t-edge-coloring of a graph G is a function that assigns one color from {1,2,…,t} to each edge. Such a coloring is called proper if no two incident edges receive the same color, and complete if every pair of distinct colors appears on a pair of incident edges. The achromatic index, denoted by α_2(G), is the largest integer t for which G admits a proper and complete t-edge-coloring. We establish new lower and upper bounds for α_2(J(n,2)), provide explicit proper and complete edge-colorings attaining the lower bounds, and determine the exact value of α_2(J(n,2)) for several values of n.

In this paper, we construct two infinite families of graphs \(G(d,c)\) and \(G^+(d,c)\), where, in both cases, a vertex label is \(x_1x_2\ldots x_c\) with \(x_i\in\{1,2,\ldots, d\}\). We provide a tight lower bound on the metric dimension of \(G^+(d, c)\). Moreover, we give the definition and properties of the supertoken graphs, a generalization of the well-known token graphs. Finally, we provide an upper bound on the metric dimension of supertoken graphs.

Given a graph G = (V, E) on n vertices and an integer k between 1 and n-1, the k-token graph Fk(G) has vertices representing the k-subsets of V, and two vertices are adjacent if their symmetric difference is the two end-vertices of an edge in E. Using the theory of Markov chains of random walks and the interchange process, it was proved that the algebraic connectivities (second smallest Laplacian eigenvalues) of G and Fk(G) coincide, but a combinatorial/algebraic proof has been shown elusive. In this paper, we use the latter approach and prove that such equality holds for different new classes of graphs under perturbations, such as extended cycles, extended complete bipartite graphs, kite graphs, and graphs with a cut clique. Kite graphs are formed by a graph (head) with several paths (tail) rooted at the same vertex and with exciting properties. For instance, we show that the different eigenvalues of a kite graph are also eigenvalues of its perturbed graph obtained by adding edges. Moreover, as a particular case of one of our theorems, we generalize a recent result of Barik and Verma (2024) about graphs with a cut vertex of degree n-1. Along the way, we give conditions under which the perturbed graph G + uv, with uv is an element of E, has the same algebraic connectivity as G. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

The k-token graph F-k (G) of a graph G on n vertices is the graph whose vertices are the ((n)(k)) k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is known that the algebraic connectivity (or second Laplacian eigenvalue) of F-k (G) equals the algebraic connectivity alpha(G) of G. In this paper, we give some bounds on the (Laplacian) eigenvalues of the k-token graph (including the algebraic connectivity) in terms of the h-token graph, with h <= k. For instance, we prove that if lambda is an eigenvalue of F-k (G), but not of G, then lambda >= k alpha (G)- k + 1. As a consequence, we conclude that if alpha (G) >= k, then alpha(F-h (G)) = alpha(G) for every h <= k. (c) 2024 Published by Elsevier B.V.

We introduce the concept of a k-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its k-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.

A bipartite graph G = ( V , E ) $G=(V,E)$ with V = V 1 boolean OR V 2 $V={V}_{1}\cup {V}_{2}$ is biregular if all the vertices of each stable set, V 1 ${V}_{1}$ and V 2 ${V}_{2}$, have the same degree, r $r$ and s $s$, respectively. This paper studies difference sets derived from both Abelian and non-Abelian groups. From them, we propose some constructions of bipartite biregular graphs with diameter d = 3 $d=3$ and asymptotically optimal order for given degrees r $r$ and s $s$, meaning that asymptotically the order approaches a fixed multiple of the Moore bound. Moreover, we find some biMoore graphs, that is, bipartite biregular graphs that attain the Moore bound.

The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we propose a general method to find the spectrum and eigenspaces of the $k$-token graph $F_k(C_n)$ of a cycle $C_n$. The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of $k=2$, we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of $C_n$.

In this note, we introduce the concept of factored lift, associated with a combined voltage graph, as a generalization of the lift graph. We present a new method for computing the eigenvalues and eigenspaces of factored lifts.

A bipartite graph G=(V,E) with V=V_1∪ V_2 is biregular if all the vertices of each stable set, V_1 and V_2, have the same degree, r and s, respectively. This paper studies difference sets derived from both Abelian and non-Abelian groups. From them, we propose some constructions of bipartite biregular graphs with diameter d=3 and asymptotically optimal order for given degrees r and s. Moreover, we find some biMoore graphs, that is, bipartite biregular graphs that attain the Moore bound.

Almost Moore mixed graphs\/} appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter $2$. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters.

We consider lifting eigenvalues and eigenvectors of graphs to their factored lifts, derived by means of a combined voltage assignment in a group. The latter extends the concept of (ordinary) voltage assignments known from regular coverings and corresponds to the cases of generalized covers of Potočnik and Toledo (2021) in which a group of automorphisms of a lift acts freely on its arc set. With the help of group representations and certain matrices over complex group rings associated with the graphs to be lifted, we develop a method for the determination of the complete spectra of the factored lift graphs and derive a sufficient condition for lifting eigenvectors.

This paper describes a general method for representing $k$-token graphs of Cayley graphs as lifts of voltage graphs. This allows us to construct line graphs of circulant graphs and Johnson graphs as lift graphs on cyclic groups. As an application of the method, we derive the spectra of the considered token graphs. The method can also be applied to dealing with other matrices, such as the Laplacian or the signless Laplacian, and to construct token digraphs of Cayley digraphs.

An (r,z,k)-mixed graph G has every vertex with undirected degree r, directed in- and out-degree z, and diameter k. In this paper, we study the case r=z=1, proposing some new constructions of (1,1,k)-mixed graphs with a large number of vertices N. Our study is based on computer techniques for small values of k and the use of graphs on alphabets for general k. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of (1,1,k)-mixed graphs are proposed with diameter of the order of 2log2N.

The k-token graph F_k(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of F_k(G) equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of F_k(G) equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.