A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such ...

Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

An injective map from the vertex set of a graph G—its order may not be finite—to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is c...

The Berge–Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings. In this paper, a useful technical lemma is proved that a cubic graph G admits a Berge–Fulkerson coloring if and only if the graph G contains a pair of edge-disjoint matchings M1 and M2 such that (i) M1 ∪ M2 in...

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f (u) + f (v) = f (w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K3 has at most two saturated ...

The edge tenacity Te(G) of a graph G is dened as:Te(G) = min {[|X|+τ(G-X)]/[ω(G-X)-1]|X ⊆ E(G) and ω(G-X) > 1} where the minimum is taken over every edge-cutset X that separates G into ω(G - X) components, and by τ(G - X) we denote the order of a largest component of G. The objective of this paper is to determine this quantity for split graphs. Let G = (Z; I; E) be a noncomplete connected split...

We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions ...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...

Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k + 3− 4 √ 2(k − 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G ...

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor vertices and/or edges (viewed as graph) order detect and prevent failures. Inspired by two notions studied literature (edge-geodetic sets distance-edge-monitoring sets), we define notion monitoring edge-geodetic set (MEG-set for short) graph G an $$S\subseteq V(G)$$ (that is, every ...