Let G be a finite simple graph having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. In an earlier paper we determined an upper bound for def(G) when G is regular and connected. This upper bound is in general not sharp when G is triangle free. In this paper we study the case when G is triangle free and r-regular. We present an upper bound for defC...

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

in recent years, many changes have been observed in buying culture of customers and developing chain stores in iran. moreover, need to be more competition in business environments made necessary the stablishment of chain stores, the creation of competitive advantage and long relations to the costumers. regular marketing is a response to stores requirement when establishing such a win-win long r...

let $g$ be a non-abelian group. the non-commuting graph $gamma_g$ of $g$ is defined as the graph whose vertex set is the non-central elements of $g$ and two vertices are joined if and only if they do not commute.in this paper we study some properties of $gamma_g$ and introduce $n$-regular $ac$-groups. also we then obtain a formula for szeged index of $gamma_g$ in terms of $n$, $|z(g)|$ and $|g|...

Let μ (G) and μmin (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) Let G be a regular graph of order n and diameter D. If H is a proper subgraph of G, then μ (G) − μ (H) > 1 nD . (ii) If G is a connected regular nonbipartite graph of order n and diameter D, then μ (G) + μmin (G) > 1 nD .

In a dispersable book embedding, the vertices of a given graphG must be ordered along a line `, called spine, and the edges of G must be drawn at different half-planes bounded by `, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding...

Let $mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $mathcal{A}^prime$. In this paper we study the quotient Arens regularity of $mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^infty(G)$. Some related result are given as well.

A graph G is called super edge-magic if there exists a bijective function f : V (G) ∪ E (G) → {1, 2, . . . , |V (G)|+ |E (G)|} such that f (V (G)) = {1, 2, . . . , |V (G)|} and f (u) + f (v) + f (uv) is a constant for each uv ∈ E (G). A graph G with isolated vertices is called pseudo super edge-magic if there exists a bijective function f : V (G) → {1, 2, . . . , |V (G)|} such that the set {f (...

We give a complete set of orthogonal invariants for triangles in G2 ( n ). As a consequence we characterize regular triangles and isoclinic triangles and we exhibit the existence regions of these objects in comparison with the angular invariants associated to them.