Simple Groups Whose Prime Graph or Solvable Graph is Split

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.

The Complexity of Power Graphs Associated With Finite Groups

The power graph $mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of a clique--replaced graph and study some applications. Second, we derive some explicit formulas concerning the complexity $kappa(mathcal{P}(G))$ for various groups $G$ such as the cyclic group of order $n$, the simple groups $L_2(q)$, the extra--special $p$--groups of order $p^3$, the Frobenius groups, etc.

The number of spanning trees of power graphs associated with specific groups and some applications

Given a group $G$, we define the power graph $mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $langle xranglesubseteq langle yrangle$ or $langle yranglesubseteq langle xrangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. In the present paper, among other results, we will find the number of spanning trees of the power graph associated with specific finite groups. We also determine, up to isomorphism, the structure of a finite group $G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we show that the alternating group $mathbb{A}_5$ is uniquely determined by tree-number of its power graph among all finite simple groups.

Some Infinite Matrices Whose Leading Principal Minors Are Well-known Sequences

There are scattered results in the literature showing that the leading principal minors of certain infinite integer matrices form the Fibonacci and Lucas sequences. In this article, among other results, we have obtained new families of infinite matrices such that the leading principal minors of them form a famous integer (sub)sequence, such as Fibonacci, Lucas, Pell and Jacobsthal (sub)sequences.

Several Quantitative Characterizations of Some Specific Groups

Let $G$ be a finite group and let $pi(G)={p_1, p_2, ldots, p_k}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${rm GK}(G)$, is defined as follows: its vertex set is $pi(G)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_ip_j$. The degree of a vertex $p_i$ in ${rm GK}(G)$ is denoted by $d_G(p_i)$ and the $k$-tuple $D(G)=left(d_G(p_1), d_G(p_2), ldots, d_G(p_k)right)$ is said to be the degree pattern of $G$. Moreover, if $omega subseteq pi(G)$ is the vertex set of a connected component of ${rm GK}(G)$, then the largest $omega$-number which divides $|G|$, is said to be an order component of ${rm GK}(G)$. We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as $U_4(2)$. Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as $U_5(2)$.