Assumed to be undirected, simple, and connected are all of the graphs in this study, and adjacency matrix A serves as the associated matrix. In this paper we show that it is possible to relate a creation sequence for a type of cographs (we call it C-graphs). Those cographs can be defined by a finite sequence of natural numbers. Using that sequence we obtain the inertia of the cograph under consideration. An extended eigenvalue-free set from (-1,0) to -1-22,-1)∪(-1,0)∪(0,-1+22αmin, (where αmin≥1 is the smallest integer of the creation sequence) is obtained for the cographs under consideration. Additionally, an exact formula is found for the characteristic polynomial.

Let G be a finite group and let P(G) be the undirected power graph of G. Recall that the vertices of P(G) are labelled by the elements of G, with an edge between g1 and g2 if either g1∈〈g2〉 or g2∈〈g1〉. The subgraph induced by the non-identity elements is called the reduced power graph, denoted by P⁎(G). The main purpose of this paper is to investigate the finite groups whose reduced power graph is claw-free, which means that it has no vertex with three pairwise non-adjacent neighbours. In particular, we prove that if P⁎(G) is claw-free, then either G is solvable or G is an almost simple group. In the second case, the socle of G is isomorphic to PSL(2,q) for suitable choices of q. Finally we prove that if P⁎(G) is claw-free, then the order of G is divisible by at most 5 different primes.

In this paper we consider particular graphs defined by (Formula Presented.), where k is even, Kα is a complete graph on α vertices, ∪ stands for the disjoint union and an overline denotes the complementary graph. These graphs do not contain the 4-vertex path as an induced subgraph, i.e., they belong to the class of cographs. In addition, they are iteratively constructed from the generating sequence (α1,α2,…,αk). Our primary question is which invariants or graph properties can be deduced from a given sequence. In this context, we compute the Lapacian eigenvalues and the corresponding eigenspaces, and derive a lower and an upper bound for the number of distinct Laplacian eigenvalues. We also determine the graphs under consideration with a fixed number of vertices that either minimize or maximize the algebraic connectivity (that is the second smallest Laplacian eigenvalue). The clique number is computed in terms of a generating sequence and a relationship between it and the algebraic connectivity is established.

We consider a connected threshold graph G with A, S as its adjacency matrix and Seidel matrix respectively. In this paper several spectral properties of S are analysed. We compute the characteristic polynomial and determinant of S. A formula for the multiplicity of the Seidel eigenvalues ±1 and characterisation of threshold graphs with at most five distinct Seidel eigenvalues are derived. Finally it is shown that two non isomorphic threshold graphs may be cospectral for S.