An incidence in a graph G is a pair (v, e) with v ∈ V (G) and e ∈ E(G), such that v and e are incident. Two incidences (v, e) and (w, f) are adjacent if v = w, or e = f, or the edge vw equals e or f. The incidence chromatic number of G is the smallest k for which there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n = Cm2Cn equals 5 when...

We prove a two-point concentration for the independent domination number of the random graph $G_{n,p}$ provided p²ln(n) ≥ 64ln((lnn)/p).

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathrm{i}r\left(G\right)=1$, $n-1$,...

By a ternary structure we mean an ordered pair $(X_0,T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0,T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if $$\{x\in X_0{T}_0(u,x,v)\}\cup \{x\in X_0{T}_0(v,x,u)\}=\{u,v\}.$$ A ternary structure $(X_0,T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u,x,y\in X_0$: $T_0(x,u,y)$ if and only if $u$ belongs to an induced $x-y$...

Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic...

The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties...

The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = ∑uv∈E(G) |dG(u) − dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and sym- metric difference. We give exact expressions or (sharp) upper bounds on the irregularity of graphs under the above mentioned...