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Register a new userOn the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs
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The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.
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Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown to be of special interest, as they are extreme in the sense of an Alon-Boppana theorem, and they have remarkable combinatorial features, such as small diameter, Chernoff bound for sampling, optimal covering time and sharp cutoff. Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alons conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan digraphs.
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the number of vertices of every reduced graph of rank r is at most $m(r)=2^{(r+2)/2}-2$ if r is even and $m(r) = 5cdot2^{(r-3)/2}-2$ if r is odd. In this article, we prove that if the conjecture is not true, then there would be a counterexample of rank at most $46$. We also show that every reduced graph of rank r has at most $8m(r)+14$ vertices.
There is a digraph corresponding to every square matrix over $mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic, and permanent polynomials can be calculated in terms of characteristic, and permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic, and permanent polynomials; the determinant, and permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.
The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyns and Temperley--Fishers ideas about spectral properties of weighted adjacency matrices of planar bipartite graphs to snake graphs. First we focus on snake graphs whose set of turning vertices is monochromatic. We provide recursive sequences to compute the characteristic polynomials; they are indexed by the upper or the lower boundary of the graph and are determined by a neighbour count. As an application, we compute the characteristic polynomials for L-shaped snake graphs and staircases in terms of Fibonacci product polynomials. Next, we introduce a method to compute the characteristic polynomials as convergents of continued fractions. Finally, we show how to transform a snake graph with turning vertices of two colours into a graph with the same number of perfect matchings to which we can apply the results above.
The cut-rank of a set $X$ in a graph $G$ is the rank of the $Xtimes (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $lvert Xrvert$ or $lvert V(G)-Xrvert $ is less than $k+ell$. We prove that every prime $3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime $3^{+3}$-rank-connected pivot-minor $H$ such that $lvert V(H)rvert =lvert V(G)rvert -1$. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most $k$ has at most $(3.5 cdot 6^{k}-1)/5$ vertices for $kge 2$. We also show that the excluded pivot-minors for the class of graphs of rank-width at most $2$ have at most $16$ vertices.
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