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Register a new userUndirected determinant, permanent and their complexity
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We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. The concept of the undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.
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Let $G$ be a graph(directed or undirected) having $k$ number of blocks. A $mathcal{B}$-partition of $G$ is a partition into $k$ vertex-disjoint subgraph $(hat{B_1},hat{B_1},hdots,hat{B_k})$ such that $hat{B}_i$ is induced subgraph of $B_i$ for $i=1,2,hdots,k.$ The terms $prod_{i=1}^{k}det(hat{B}_i), prod_{i=1}^{k}text{per}(hat{B}_i)$ are det-summands and per-summands, respectively, corresponding to the $mathcal{B}$-partition. The determinant and permanent of a graph having no loops on its cut-vertices is equal to summation of det-summands and per-summands, respectively, corresponding to all possible $mathcal{B}$-partitions. Thus, in this paper we calculate determinant and permanent of some graphs, which include block graph with negatives cliques, signed unicyclic graph, mix complete graph, negative mix complete graph, and star mix block graphs.
Let $D$ be a strongly connected digraph. The average distance $bar{sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $rho(D)$ and proximity $pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $pi(D)$ and $rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $pi(T)=rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $pi(D)=rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $pi(D)=rho(D).$ We describe such a family for undirected graphs as well.
We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the emph{rectangular} symbolic matrix in both commutative and noncommutative settings. The main results are: 1. We show an explicit $O^{*}({nchoose {downarrow k/2}})$-size ABP construction for noncommutative permanent polynomial of $ktimes n$ symbolic matrix. We obtain this via an explicit ABP construction of size $O^{*}({nchoose {downarrow k/2}})$ for $S_{n,k}^*$, noncommutative symmetrized version of the elementary symmetric polynomial $S_{n,k}$. 2. We obtain an explicit $O^{*}(2^k)$-size ABP construction for the commutative rectangular determinant polynomial of the $ktimes n$ symbolic matrix. 3. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is $W[1]$-hard.
By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors.
The determinants of ${pm 1}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamards maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.
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