In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(\! -\lambda ^*, -2)$, where $\lambda ^* = ho ^{1/2} + ho ^{-1/2} \approx 2.01980$, and $ho$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(\! -\lambda , -2)$ for any constant $\lambda \gt 2$.

In this article, we present a unified approach for proving several Turán-type and generalized Turán-type problems, degree power problems, and extremal spectra problems on paths, cycles, and matchings. Specifically, we generalize classical results on cycles and matchings by Kopylov, Erdős–Gallai, and Luo et al., respectively, and provide a positive resolution to an open problem originally proposed by Nikiforov. Moreover, we improve the spectral extremal results on paths, building on the work of Nikiforov, and Nikiforov and Yuan. Additionally, we provide a comprehensive solution to the connected version of the problem related to the degree power sum of a graph that contains no path on k vertices, a topic initially investigated by Caro and Yuster.

If G is a graph, then $X\subseteq V(G)$ is a general position set if for every two vertices $v,u\in X$ and every shortest $(u,v)$-path P, no inner vertex of P lies in X. We propose three algorithms to compute a largest general position set in G: an integer linear programming algorithm, a genetic algorithm and a simulated annealing algorithm. These approaches are supported by examples from different areas of graph theory.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

We show that the twin-width of every $n$-vertex $d$-regular graph is at most $n^{\frac{d-2}{2d-2}+o(1)}$ for any fixed integer $d \geq 2$ and that almost all $d$-regular graphs attain this bound. More generally, we obtain bounds on the twin-width of sparse Erdős–Renyi and regular random graphs, complementing the bounds in the denser regime due to Ahn, Chakraborti, Hendrey, Kim, and Oum.

Let G be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited while maintaining the general position property at all times. The mobile general position number of G is the cardinality of a largest mobile general position set of G. We give bounds on the mobile general position number and determine exact values for certain common classes of graphs, including block graphs, rooted products, unicyclic graphs, Kneser graphs $K(n,2)$ and line graphs of complete graphs.

All finite simple self 2-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self 2-distance graph.

The prime vertex graph, $\Delta \left( X \right)$ , and the common divisor graph, $\Gamma \left( X \right)$ , are two graphs that have been defined on a set of positive integers $X$ . Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

Let G be a connected simple graph. The degree distance of G is defined as D′(G)=∑ u∈V (G)dG(u)DG(u), where DG(u) is the sum of distances between the vertex u and all other vertices in G and dG(u) denotes the degree of vertex u in G. In contrast to many established results on extremal properties of degree distance, few results in the literature deal with the degree distance of composite graphs. Towards closing this gap, we study the degree distance of some composite graphs here. We present explicit formulas for D′ (G)of three composite graphs, namely, double graphs, extended double covers and edge copied graphs.