On chordality of the power graph of finite groups

A graph is called chordal if it forbids induced cycles of length 4 or more. In this paper, we attempt to identify the non-nilpotent groups whose power graph is a chordal graph (this question was raised by Cameron in [4]). In this direction, we characterise the direct product of finite groups having chordal power graphs. We classify all finite simple groups of Lie type whose power graph is chordal. Further, we prove that the power graph of a sporadic simple group is always non-chordal. In addition, we show that almost all groups of order up to 47 have chordal power graphs

On finite groups whose power graph is a cograph

Funding: The author Pallabi Manna is supported by CSIR (Grant No-09/983(0037)/2019-EMR-I). Ranjit Mehatari thanks the SERB, India, for financial support (File Number: CRG/2020/000447) through the Core Research Grant.A P4-free graph is called a cograph. In this paper we partially characterize finite groups whose power graph is a cograph. As we will see, this problem is a generalization of the determination of groups in which every element has prime power order, first raised by Graham Higman in 1957 and fully solved very recently. First we determine all groups G and H for which the power power graph of G times H is a cograph. We show that groups whose power graph is a cograph can be characterised by a condition only involving elements whose orders are prime or the product of two (possibly equal) primes. Some important graph classes are also taken under consideration. For finite simple groups we show that in most of the cases their power graphs are not cographs: the only ones for which the power graphs are cographs are certain groups PSL(2,q) and Sz(q) and the group PSL(3,4). However, a complete determination of these groups involves some hard number-theoretic problems.PostprintPeer reviewe

Forbidden subgraphs of power graphs

Funding: CSIR, India (Grant No-09/983(0037)/2019-EMR-I), SERB, India through Core Research Grant (File Number-CRG/2020/000447).The undirected power graph (or simply power graph) of a group G, denoted by P(G), is a graph whose vertices are the elements of the group G, in which two vertices u and v are connected by an edge between if and only if either u = vi or v = uj for some i,j. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups G the power graph P(G) lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.Publisher PDFPeer reviewe

On finite groups whose power graph is a cograph

Funding: The author Pallabi Manna is supported by CSIR (Grant No-09/983(0037)/2019-EMR-I). Ranjit Mehatari thanks the SERB, India, for financial support (File Number: CRG/2020/000447) through the Core Research Grant.A P4-free graph is called a cograph. In this paper we partially characterize finite groups whose power graph is a cograph. As we will see, this problem is a generalization of the determination of groups in which every element has prime power order, first raised by Graham Higman in 1957 and fully solved very recently. First we determine all groups G and H for which the power power graph of G times H is a cograph. We show that groups whose power graph is a cograph can be characterised by a condition only involving elements whose orders are prime or the product of two (possibly equal) primes. Some important graph classes are also taken under consideration. For finite simple groups we show that in most of the cases their power graphs are not cographs: the only ones for which the power graphs are cographs are certain groups PSL(2,q) and Sz(q) and the group PSL(3,4). However, a complete determination of these groups involves some hard number-theoretic problems.PostprintPeer reviewe

Eigenvalue bounds for some classes of matrices associated with graphs

summary:For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first, we derive bounds for the second largest and the smallest eigenvalues of adjacency matrices of $k$-regular graphs. Then we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest and the largest eigenvalues of the Laplacian matrices of graphs. The sharpness of these bounds is verified by examples

Forbidden subgraphs of power graphs

The undirected power graph (or simply power graph) of a group G, denoted by P(G), is a graph whose vertices are the elements of the group G, in which two vertices u and v are connected by an edge between if and only if either u = vi or v = uj for some i,j.A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups G the power graph P(G) lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems