We introduce a combinatorial shifting operation on multicomplexes that carries similar properties required for the ordinary shifting operation on simplicial complexes. A linearly colored simplicial complex is called shifted if its associated multicomplex is stable under defined operation. We show that the underlying simplicial subcomplex of a linearly shifted simplicial complex is shifted in the ordinary sense, while the ordinary and linear shiftings are not interrelated in general. Separately, we also prove that any linearly shifted complex must be shellable with respect to the order of its facets induced by the linear coloring. As an application, we provide a characterization of simple graphs whose independence complexes are linearly shifted. The class of graphs obtained constitutes a superclass of threshold graphs.

A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.

Explicit expressions for all the primitive idempotents in the ring \(R_{2^n} = {F}_q[x]/(x^{2^n} – 1)\), where \(q\) is an odd prime power, are obtained. Some lower bounds on the minimum distances of the irreducible cyclic codes of length \(2^n\) over \({F}_q\) are also obtained.

In this note we prove that all connected Cayley graphs of every finite group \(Q \times H\) are \(1\)-factorizable, where \(Q\) is any non-trivial group of \(2\)-power order and \(H\) is any group of odd order.

A graph \(G\) is called super vertex-magic total labelings if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(v) + \sum_{u \sim v} f(vu) = C\), where the sum is over all vertices \(u\) adjacent to \(v\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\), \(f(E(G)) = \{|V(G)|+1,|V(G)|+2,\ldots,|V(G)|+|E(G)|\}\). \({The Knödel graphs}\) \(W_{\Delta,n}\) have even \(n \geq 2\) vertices and degree \(\Delta\), \(1 \leq \Delta \leq \lfloor\log_2 n\rfloor\). The vertices of \(W_{\Delta,n}\) are the pairs \((i,j)\) with \(i = 1,2\) and \(0 \leq i \leq n/2-1\). For every \(j\), \(0 \leq j \leq n/2-1\), there is an edge between vertex \((1,j)\) and every vertex \((2,(j+2^k-1) \mod (n/2))\), for \(k=0,\ldots,\Delta-1\). In this paper, we show that \(W_{3,n}\) is super vertex-magic for \(n \equiv 0 \mod 4\).

Evolutionary graphs were initially proposed by Lieberman \(et \;al\). and evolutionary dynamics on two levels are recently introduced by Traulsen et al. We now introduce a new type of evolutionary dynamics,evolutionary graphs on two levels, and the fixation probability is analyzed. Some interesting results, evolutionary graphs on two levels are more stable than single level evolutionary graphs, are obtained in this paper.

A vertex \(k\)-ranking of a graph \(G\) is a function \(c: V(G) \to \{1,\ldots,k\}\) such that if \(c(u) = c(v)\), \(u,v \in V(G)\), then each path connecting vertices \(u\) and \(v\) contains a vertex \(w\) with \(c(w) > c(u)\). If each vertex \(v\) has a list of integers \(L(v)\) and for a vertex ranking \(c\) it holds \(c(v) \in L(v)\) for each \(v \in V(G)\), then \(c\) is called an \(L\)-list \(k\)-ranking, where \(\mathcal{L} = \{L(v) : v \in V(G)\}\). In this paper, we investigate both vertex and edge (vertex ranking of a line graph) list ranking problems. We prove that both problems are NP-complete for several classes of acyclic graphs, like full binary trees, trees with diameter at most \(4\), and comets. The problem of finding vertex (edge) \(\mathcal{L}\)-list ranking is polynomially solvable for paths and trees with a bounded number of non-leaves, which includes trees with diameter less than \(4\).

In this paper we determine unique graph with largest spectral radius among all tricyclic graphs with \(n\) vertices and \(k\) pendant edges.

A new proof is given to the following result of ours. Let \(G\) be an outerplanar graph with maximum degree \(\Delta \geq 3\). The chromatic number \(\chi(G^2)\) of the square of \(G\) is at most \(\Delta+2\), and \(\chi(G^2) = \Delta+1\) if \(\Delta \geq 7\).