We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard’s conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S5, S6, and S7 for involutions.

Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α + β)(m) = α(m) + β(m), (α ◦ β)(m) = α(β(m)). We must verify the four subring conditions from 3.2.2. (i) The additive identity of End(M) is z : M → M, m → 0M . Note that for any m,n ∈ M z(m + n) = 0M ...

A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G...

We study certain quotients of generalized Artin groups which have a natural map onto D-type Artin groups, where the generalized Artin group A(T ) is defined by a signed graph T . Then we find a certain quotient G(T ) according to the graph T , which also have a natural map onto A(Dn). We prove that G(T ) is isomorphic to a semidirect product of a group K(m,n), with the Artin group A(Dn), where ...

Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a par...

A network of n sensors receiving independent and identical observations in R", regarding certain hinarj hypotheses, pass their decisions to a fusion center which then decides which one of the two hypotheses is true. We consider the situation where each sensor employs a likelihood ratio test with its own observation and a threshold, which is the same for all the sensors, and the fusion center de...

The normalized Laplacian of a graph was introduced by F.R.K. Chung and has been studied extensively over the last decade. In this paper, we introduce the notion of the normalized Laplacian of signed graphs and extend some fundamental concepts of the normalized Laplacian from graphs to signed graphs.

An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families – balanced, balanceable, and unbalanceable – and we obtain a complete classificati...

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arcset $A(D)$. A twin signed total Roman dominating function (TSTRDF) on thedigraph $D$ is a function $f:V(D)rightarrow{-1,1,2}$ satisfyingthe conditions that (i) $sum_{xin N^-(v)}f(x)ge 1$ and$sum_{xin N^+(v)}f(x)ge 1$ for each $vin V(D)$, where $N^-(v)$(resp. $N^+(v)$) consists of all in-neighbors (resp.out-neighbors) of $v$, and (...

A signed graph is a graph whose edges are signed. In a vertex-signed graph the vertices are signed. The latter is called consistent if the product of signs in every circle is positive. The line graph of a signed graph is naturally vertexsigned. Based on a characterization by Acharya, Acharya, and Sinha in 2009, we give constructions for the signed simple graphs whose naturally vertex-signed lin...