You wrote:
"The data you have on my great-grandfather, Elihu C. Graham, matches what I have but I am very surprised to learn his brother M.C. was listed as deserting. My data does show him enlisting with his brothers in March, 1862. I do not have any muster roll indicating his being detailed as a teamster carrying ordnance. Does your muster roll indicate the location and date of this duty? More importantly, in whose unit and where was it located at the time of his desertion? I do have only one muster roll and it's apparently the same one you are referencing about his unit surrendering on 11 May 1865."

you are looking at his one reference as a member od Co A Davies BN, all the other information is in the records of the 25th Ark Inf. He was detailed as a teamster on Sept 1, 1862. He is present with the regiment again Nov 62-Feb 63 when it shows him driving ordnance wagons in March and April 1863. He was still in 25th Ark Inf and his brigade was probably in Mississippi by that time but no large battle as such. many men did indeed go home which technically means deserted, but they then joined a unit back in Ark. Especially after Vicksburg became surrounded men went home before the Mississippi River was totally controlled by US forces. So, he merely left one unit (Co D 25th Ark Inf) for another (Co A Davies Ark Cav BN), so just depends on how one wants to look at it. He remained in service so he was still a soldier, did not try to get out of it so to speak

You wrote:
"I really question the "desertion." First, that last muster & descriptive roll states it's "of prisoners of war belonging to the Army of the Northern Sub-District of Arkansas." Could it be that he was captured rather than deserting on June 26, 1863? Do you know if his unit was engaged in battle on that date? If he had deserted in 1863, then re-joined his unit 2 years later for the surrender, wouldn't his commander have immediately taken him as a prisoner for the Confederacy? Hard to make sense of it."

As stated above, he deserted/left the 25th Ark Inf and then joined a new unit in Ark, Co A Davies Ark Cav BN which is the unit he surrendered with at the end.

You wrote:
"Further, M.C. applied for a Confederate pension in 1899 while living in Texas. Of course, he had multiple fellow soldiers to support his application swearing that he served throughout the war, never deserting. On submission, the comptroller verified his service record without any reference to desertion. His application was "denied" but it was because he had sold some farm land to a son prior to asking for a pension so it was seen as trying to get rid of assets so as to be eligible for pension."

Yes, there are quite a few men who actually got pensions who at some point on their current CSR's state they deserted. But again, they may have continued to serve so weren't true deserters. Also to remember is the fact that the men signing for them, unless the soldier left and turned to the enemy for support or actually supported the enemy, did not necessarily consider them a deserter for going home to take care of family. They might overlook the "self imposed furlough" in order to assist a fellow Confederate to get one over on the govt and help his situation with a small pension.

You wrote:
"Could you please explain further about John C. Graham as I have only that he enlisted with his brothers in March 1862 as 2 Lt. I don't have any document indicating a promotion from 3rd Lt. or about a unit reorganization at Corinth. Do you have a muster roll indicating he was not "retained"? What exactly does that mean as I have never heard of that? Could he just have decided not to re-enlist and returned home?"

In May of 1862 the Confederate Army undewent a huge reorganization after the battle of Shiloh. Officers were elected by the men and after the first months of the war, many were not as popular with the men and when reorganized, they could "fire" them and elect new officers. Or an officer could decide he did not want to stand for reelection. That is evidently what happened to John Graham. He enlisted and was made 3LT, he was elevated to 2nd LT but then in May 1862 was not chosen or he chose not to run for reelction, therefore my comment "not retained". It could have been either one

All of this information comes right off his CSR papers which are from the National Archives in DC. Again, not sure if he was the same man in McGee's ARk Cav but its a good possibility.