Let G be a graph. The first Zagreb M1(G) of graph G is defined as: M1(G) = uV(G) deg(u)2. In this paper, we prove that each even number except 4 and 8 is a first Zagreb index of a caterpillar. Also, we show that the fist Zagreb index cannot be an odd number. Moreover, we obtain the fist Zagreb index of some graph operations.

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In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular graphs. Our technique involves semidefinite programming and exploiting the algebra structure of the optimization problem so that the question of finding a lowe...

Consider the problem of using a vehicle to transport k objects one at a time between s stations on a circular track. Let the cost of the transportation be the rotal distance traveled by the vehicle on the track. An 0 (k + M(s, q)) time algorithm is presented to find a minimum cost transportation, where M (m, n) is the time to solve a minimum spanning tree problem on a graph with m edges and n v...

In [M. Klin, A. Munemusa, M. Muzychuk, P.-H. Zieschang Directed strongly regular graphs obtained from coherent algebras. Linear Algebra and its Applications 337, (2004) 83-109] the flag algebra of a given balanced incomplete block design with parameters (ν, b, r, k, λ) where λ = 1, has been constructed. In this paper, we consider the association scheme which is related to this flag algebra. By ...

It is shown that the graphs for which the Szeged index equals ‖G‖·|G| 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables to disprove a conjecture proposed in [Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of ...

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We sh...