A note on Hoffman-type identities of graphs

A note on Hoffman-type identities of graphs

An eigenvalue of a graph G is called main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Hoffman [A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963) 30–36] proved that G is a connected k-regular graph if and only if n ∏t i=2(A− λiI ) = ∏t i=2(k − λi) · J , where I is the unit matrix and J the all-one matrix and λ1 = k, λ2, . . . , λt ar...

A note on vague graphs

In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.

A Note on Tensor Product of Graphs

Let $G$ and $H$ be graphs. The tensor product $Gotimes H$ of $G$ and $H$ has vertex set $V(Gotimes H)=V(G)times V(H)$ and edge set $E(Gotimes H)={(a,b)(c,d)| acin E(G):: and:: bdin E(H)}$. In this paper, some results on this product are obtained by which it is possible to compute the Wiener and Hyper Wiener indices of $K_n otimes G$.

A Note on Strongly Quotient Graphs

A note on 3-Prime cordial graphs

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....